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Gdynia

Gdynia (Gdiniô; Gdingen, Gotenhafen ) is a city in northern Poland and a seaport on the Baltic Sea coast. With an estimated population of 257 000, it is the 12th-largest city in Poland and the second-largest in the Pomeranian Voivodeship after Gdańsk. Gdynia is part of a conurbation with the spa town of Sopot, the city of Gdańsk, and suburban communities, which together form a metropolitan area called the Tricity (Trójmiasto) with around one million inhabitants.

Historically and culturally part of Kashubia and Eastern Pomerania, Gdynia for centuries remained a small fishing village. By the 20th-century it attracted visitors as a seaside resort town. In 1926, Gdynia was granted city rights after which it enjoyed demographic and urban development, with a modernist cityscape. It became a major seaport city of Poland. In 1970, protests in and around Gdynia contributed to the rise of the Solidarity movement in nearby Gdańsk.

The port of Gdynia is a regular stopover on the cruising itinerary of luxury passenger ships and ferries travelling to Scandinavia. Gdynia's downtown, designated a historical monument of Poland in 2015, is an example of building an integrated European community and includes Functionalist architectural forms. It is also a candidate for the UNESCO World Heritage List. Its axis is based around 10 Lutego Street and connects the main train station with the Southern Pier. The city is also known for holding the annual Gdynia Film Festival. In 2013, Gdynia was ranked by readers of The News as Poland's best city to live in, and topped the national rankings in the category of "general quality of life". In 2021, the city entered the UNESCO Creative Cities Network and was named UNESCO City of Film.

Early history
The area of the later city of Gdynia shared its history with Pomerelia (Eastern Pomerania). In prehistoric times, it was the center of Oksywie culture; it was later populated by Slavs with some Baltic Prussian influences. In the late 10th century, the region was united with the emerging state of Poland by its first historic ruler Mieszko I. During the reign of Bolesław II, the region seceded from Poland and became independent, to be reunited with Poland in 1116/1121 by Bolesław III. In 1209, the present-day district of Oksywie was first mentioned (Oxhöft). Following the fragmentation of Poland, the region became part of the Duchy of Pomerania (Eastern), which became separate from Poland in 1227, to be reunited in 1282. The first known mention of the name "Gdynia", as a Pomeranian (Kashubian) fishing village dates back to 1253. The first church on this part of the Baltic Sea coast was built there. In 1309–1310, the Teutonic Order invaded and annexed the region from Poland. In 1380, the owner of the village which became Gdynia, Peter from Rusocin, gave the village to the Cistercian Order. In 1382, Gdynia became property of the Cistercian abbey in Oliwa. In 1454, King Casimir IV Jagiellon signed the act of reincorporation of the region to the Kingdom of Poland, and the Thirteen Years' War, the longest of all Polish-Teutonic wars, started. It ended in 1466, when the Teutonic Knights recognized the region as part of Poland. Administratively, Gdynia was located in the Pomeranian Voivodeship in the province of Royal Prussia in the Greater Poland Province of the Kingdom of Poland and later of the Polish–Lithuanian Commonwealth. The present-day neighbourhood of Kolibki was the location of the Kolibki estate, purchased by King John III Sobieski in 1685.

In 1772, Gdynia was annexed by the Kingdom of Prussia in the First Partition of Poland. Gdynia, under the Germanized name Gdingen, was included within the newly formed province of West Prussia and was expropriated from the Cistercian Order. In 1789, there were only 21 houses in Gdynia. Around that time Gdynia was so small that it was not marked on many maps of the period: it was about halfway from Oksywie and Mały Kack, now districts of Gdynia. In 1871, the village became part of the German Empire. In the early 20th century Gdynia was not a poor fishing village as it is sometimes described; it had become a popular tourist spot with several guest houses, restaurants, cafés, several brick houses and a small harbour with a pier for small trading ships. The first Kashubian mayor was Jan Radtke. It is estimated that around 1910 the population of Gdynia was 895 people.

Following World War I, in 1918, Poland regained independence, and following the Treaty of Versailles, in 1920, Gdynia was re-integrated with the reborn Polish state. Simultaneously, the nearby city of Gdańsk (Danzig) and surrounding area was declared a free city and put under the League of Nations, though Poland was given economic liberties and requisitioned for matters of foreign representation.

Construction of the seaport
The decision to build a major seaport at Gdynia village was made by the Polish government in winter 1920, in the midst of the Polish–Soviet War (1919–1920). The authorities and seaport workers of the Free City of Danzig felt Poland's economic rights in the city were being misappropriated to help fight the war. German dockworkers went on strike, refusing to unload shipments of military supplies sent from the West to aid the Polish army, and Poland realized the need for a port city it was in complete control of, economically and politically.

Construction of Gdynia seaport started in 1921 but, because of financial difficulties, it was conducted slowly and with interruptions. It was accelerated after the Sejm (Polish parliament) passed the Gdynia Seaport Construction Act on 23 September 1922. By 1923 a 550-metre pier, 175 m of a wooden tide breaker, and a small harbour had been constructed. Ceremonial inauguration of Gdynia as a temporary military port and fishers' shelter took place on 23 April 1923. The first major seagoing ship, the French Line steamer Kentucky, arrived on 13 August 1923 after being diverted because of a strike at Gdansk. To speed up the construction works, the Polish government in November 1924 signed a contract with the French-Polish Consortium for Gdynia Seaport Construction. By the end of 1925, they had built a small seven-metre-deep harbour, the south pier, part of the north pier, a railway, and had ordered the trans-shipment equipment. The works were going more slowly than expected, however. They accelerated only after May 1926, because of an increase in Polish exports by sea, economic prosperity, the outbreak of the German–Polish trade war which reverted most Polish international trade to sea routes, and thanks to the personal engagement of Eugeniusz Kwiatkowski, Polish Minister of Industry and Trade (also responsible for the construction of Centralny Okręg Przemysłowy). By the end of 1930 docks, piers, breakwaters, and many auxiliary and industrial installations were constructed (such as depots, trans-shipment equipment, and a rice processing factory) or started (such as a large cold store).

Trans-shipments rose from 10,000 tons (1924) to 2,923,000 tons (1929). At this time Gdynia was the only transit and special seaport designed for coal exports.

In the years 1931–1939 Gdynia harbour was further extended to become a universal seaport. In 1938 Gdynia was the largest and most modern seaport on the Baltic Sea, as well as the tenth biggest in Europe. The trans-shipments rose to 8.7 million tons, which was 46% of Polish foreign trade. In 1938 the Gdynia shipyard started to build its first full-sea ship, the Olza.

Construction of the city
The city was constructed later than the seaport. In 1925 a special committee was inaugurated to build the city; city expansion plans were designed and city rights were granted in 1926, and tax privileges were granted for investors in 1927. The city started to grow significantly after 1928.

A new railway station and the Post Office were completed. The State railways extended their lines, built bridges and also constructed a group of houses for their employees. Within a few years houses were built along some 10 mi of road leading northward from the Free City of Danzig to Gdynia and beyond. Public institutions and private employers helped their staff to build houses. In 1933 a plan of development providing for a population of 250,000 was worked out by a special commission appointed by a government committee, in collaboration with the municipal authorities. By 1939 the population had grown to over 120,000.

Gdynia during World War II (1939–1945)
During the German invasion of Poland, which started World War II in September 1939, Gdynia was the site of fierce Polish defense. On 13 September 1939, the Germans carried out first arrests of local Poles in the southern part of the city, while the Polish defense was still ongoing in the northern part. On 14 September 1939, the Germans captured the entire city, and then occupied it until 1945. On 15–16 September, the Germans carried out further mass arrests of 7,000 Poles, while Polish soldiers still fought in nearby Kępa Oksywska. The German police surrounded the city and carried out mass searches of weapons. Arrested Poles were held and interrogated in churches, cinemas and halls, and then around 3,000 people were released until 18 September. The occupiers established several prisons and camps for Polish people, who were afterwards either deported to concentration camps or executed. Some Poles from Gdynia were executed by the Germans near Starogard Gdański in September 1939. In October and November 1939, the Germans carried out public executions of 52 Poles, including activists, bank directors and priests, in various parts of the city. In November 1939, the occupiers also murdered hundreds of Poles from Gdynia during the massacres in Piaśnica committed nearby as part of the Intelligenzaktion. Among the victims were policemen, officials, civil defenders of Gdynia, judges, court employees, the director and employees of the National Bank of Poland, merchants, priests, school principals, teachers, and students of local high schools. On the night of 10–11 November, the German security police carried out mass arrests of over 1,500 Poles in the Obłuże district, and then murdered 23 young men aged 16–20, in retaliation for breaking windows at the headquarters of the German security police. On 11 November, a German gendarme shot and killed two Polish boys who were collecting Polish books from the street, which were thrown out of the windows by new German settlers in the Oksywie district. The Germans renamed the city to Gotenhafen after the Goths, an ancient Germanic tribe, who had lived in the area. 10 Poles from Gdynia were also murdered by the Russians in the large Katyn massacre in April–May 1940.

Some 50,000 Polish citizens were expelled to the General Government (German-occupied central Poland) to make space for new German settlers in accordance with the Lebensraum policy. Local Kashubians who were suspected to support the Polish cause, particularly those with higher education, were also arrested and executed. The German gauleiter Albert Forster considered Kashubians of "low value" and did not support any attempts to create a Kashubian nationality. Despite such circumstances, local Poles, including Kashubians, organized Polish resistance groups, Kashubian Griffin (later Pomeranian Griffin), the exiled "Związek Pomorski" in the United Kingdom, and local units of the Home Army, Service for Poland's Victory and Gray Ranks. Activities included distribution of underground Polish press, smuggling data on German persecution of Poles and Jews to Western Europe, sabotage actions, espionage of the local German industry, and facilitating escapes of endangered Polish resistance members and British and French prisoners of war who fled from German POW camps via the city's port to neutral Sweden. The Gestapo cracked down on the Polish resistance several times, with the Poles either killed or deported to the Stutthof and Ravensbrück concentration camps. In 1943, local Poles managed to save some kidnapped Polish children from the Zamość region, by buying them from the Germans at the local train station. The harbour was transformed into a German naval base. The shipyard was expanded in 1940 and became a branch of the Kiel shipyard (Deutsche Werke Kiel A.G.). The city became an important base, due to its being relatively distant from the war theater, and many German large ships—battleships and heavy cruisers—were anchored there. During 1942, Dr Joseph Goebbels authorized relocation of SS Cap Arcona (1927) to Gotenhafen Harbour as a stand-in for RMS Titanic during filming of the German-produced movie Titanic, directed by Herbert Selpin.

The Germans set up an Einsatzgruppen-operated penal camp in the Grabówek district, a transit camp for Allied marine POWs, a forced labour subcamp of the Stalag XX-B POW camp for several hundred Allied POWs at the shipyard, and two subcamps of the Stutthof concentration camp, the first located in the Orłowo district in 1941–1942, the second, named Gotenhafen, located at the shipyard in 1944–1945.

The seaport and the shipyard both witnessed several air raids by the Allies from 1943 onwards, but suffered little damage. Gdynia was used during winter 1944–45 to evacuate German troops and refugees trapped by the Red Army. Some of the ships were hit by torpedoes from Soviet submarines in the Baltic Sea on the route west. The ship MV Wilhelm Gustloff sank, taking about 9,400 people with her – the worst loss of life in a single sinking in maritime history. The seaport area was largely destroyed by withdrawing German troops and millions of encircled refugees in 1945 being bombarded by the Soviet military (90% of the buildings and equipment were destroyed) and the harbour entrance was blocked by the German battleship GERMAN BATTLESHIP Gneisenau that had been brought to Gotenhafen for major repairs.

After World War II
On 28 March 1945, the city was captured by the Soviets and restored to Poland. The Soviets installed a communist regime, which stayed in power until the Fall of Communism in the 1989. The post-war period saw an influx of settlers from Warsaw which was destroyed by Germany, and other parts of the country as well as Poles from the cities of Wilno (now Vilnius) and Lwów (now Lviv) from the Soviet-annexed former eastern Poland. Also Greeks, refugees of the Greek Civil War, settled in the city. The port of Gdynia was one of the three Polish ports through which refugees of the Greek Civil War reached Poland.

On December 17, 1970, worker demonstrations took place at Gdynia Shipyard. Workers were fired upon by the police. Janek Wiśniewski was one of 40 killed, and was commemorated in a song by Mieczysław Cholewa, Pieśń o Janku z Gdyni. One of Gdynia's important streets is named after Janek Wiśniewski. The event was also portrayed in Andrzej Wajda's movie Man of Iron.

On 4 December 1999, a storm destroyed a huge crane in a shipyard.

Climate
The climate of Gdynia is an oceanic climate owing to its position of the Baltic Sea, which moderates the temperatures, compared to the interior of Poland. The climate is mild and there is a somewhat uniform precipitation throughout the year. Autumns are significantly warmer than springs because of the warming influence of the Baltic Sea. Nights on average are warmer than in the interior of the country. Typical of Northern Europe, there is little sunshine during late autumn, winter and early spring, but plenty during late spring and summer. Because of its northerly latitude, Gdynia has 17 hours of daylight in midsummer but only around 7 hours in midwinter. The lowest pressure in Poland was recorded in Gdynia - 960.2 hPa on January 17, 1931.

Districts
Gdynia is divided into smaller divisions: dzielnicas and osiedles. Gdynia's dzielnicas include: Babie Doły, Chwarzno-Wiczlino, Chylonia, Cisowa, Dąbrowa, Działki Leśne, Grabówek, Kamienna Góra, Karwiny, Leszczynki, Mały Kack, Obłuże, Oksywie, Orłowo, Pogórze, Pustki Cisowskie-Demptowo, Redłowo, Śródmieście, Wielki Kack, Witomino-Leśniczówka, Witomino-Radiostacja, Wzgórze Św. Maksymiliana.

Osiedles: Bernadowo, Brzozowa Góra, Chwarzno, Dąbrówka, Demptowo, Dębowa Góra, Fikakowo, Gołębiewo, Kacze Buki, Kolibki, Kolonia Chwaszczyno, Kolonia Rybacka, Krykulec, Marszewo, Międzytorze, Niemotowo, Osada Kolejowa, Osada Rybacka, Osiedle Bernadowo, Port, Pustki Cisowskie, Tasza, Wiczlino, Wielka Rola, Witomino, Wysoka, Zielenisz.

Cityscape
Gdynia is a relatively modern city. Its architecture includes the 13th century St. Michael the Archangel's Church in Oksywie, the oldest building in Gdynia, and the 17th century neo-Gothic manor house located on Folwarczna Street in Orłowo.

The surrounding hills and the coastline attract many nature lovers. A leisure pier and a cliff-like coastline in Kępa Redłowska, as well as the surrounding Nature Reserve, are also popular locations. In the harbour, there are two anchored museum ships, the destroyer ORP Błyskawica and the tall ship frigate Dar Pomorza. A 1.5 km-long promenade leads from the marina in the city center, to the beach in Redłowo.

Most of Gdynia can be seen from Kamienna Góra (54 m asl) or the viewing point near Chwaszczyno. There are also two viewing towers, one at Góra Donas, the other at Kolibki.

In 2015 the Emigration Museum opened in the city. Other museums include the Gdynia Aquarium, Experyment Science Center, Abraham's house, Żeromski's house, Gdynia Automotive Museum, Naval Museum, and Gdynia City Museum.

Modernist Center
Gdynia holds many examples of early 20th-century architecture, especially monumentalism and early functionalism, and modernism. Historic Urban Layout of the City Center was drafted by Adam Kuncewicz and Roman Feliński in 1926. The central axis of Gdynia is built around 10 Lutego Street, Kosciuszka Square and the Southern Pier. The structure of the city is designed to emphasize the connection of Gdynia and Poland with the Baltic Sea. Examples of modernist architecture are the buildings of the Bank of Poland and many tenement houses (kamienice). Another good example of modernism is PLO Building situated at 10 Lutego Street.

The architecture of central Gdynia was inspired by the work of European architects such as Erich Mendelssohn and is sometimes compared to the White City of Tel Aviv. The center of Gdynia has become a symbol of modernity, but was included in the list of historical monuments of Poland and is a candidate for the UNESCO World Heritage List.

Culture
Gdynia hosts the Gdynia Film Festival, the main Polish film festival. The International Random Film Festival was hosted in Gdynia in November 2014. Since 2003 Gdynia has been hosting the Open'er Festival, one of the biggest contemporary music festivals in Europe. The festival welcomes many foreign hip-hop, rock and electronic music artists every year. In record-high 2018 it was attended by over 140,000 people, who enjoyed the lineup headlined by Bruno Mars, Gorillaz, Arctic Monkeys, and Depeche Mode. Another important summer event in Gdynia is the Viva Beach Party, which is a large two-day techno party made on Gdynia's Public Beach and a summer-welcoming concerts CudaWianki. Gdynia also hosts events for the annual Gdańsk Shakespeare Festival.

In the summer of 2014 Gdynia hosted Red Bull Air Race World Championship.

Cultural references
In 2008, Gdynia made it onto the Monopoly Here and Now World Edition board after being voted by fans through the Internet. Gdynia occupies the space traditionally held by Mediterranean Avenue, being the lowest voted city to make it onto the Monopoly Here and Now board, but also the smallest city to make it in the game. All of the other cities are large and widely known ones, the second smallest being Riga. The unexpected success of Gdynia can be attributed to a mobilization of the town's population to vote for it on the Internet.

An abandoned factory district in Gdynia was the scene for the survival series Man vs Wild, season 6, episode 12. The host, Bear Grylls, manages to escape the district after blowing up a door and crawling through miles of sewer.

Ernst Stavro Blofeld, the supervillain in the James Bond novels, was born in Gdynia on 28 May 1908, according to Thunderball.

Gdynia is sometimes called "Polish Roswell" due to the alleged UFO crash on 21 January 1959.

Notable people

 * Stanisław Baranowski (1935–1978), glaciologist, undertook scientific expeditions to Spitsbergen and Antarctica
 * Karol Olgierd Borchardt (1905–1986), writer and captain of the Polish Merchant Marine
 * Krzysztof Charamsa (born 1972), former Catholic theologian and author
 * Zbigniew Ciesielski (1934–2020), mathematician
 * Adam Darski (born 1977), musician and TV personality, frontman for the blackened death metal band Behemoth
 * Wiesław Dawidowski (born 1964), Augustinian Catholic priest, doctor of theology and journalist
 * Rafał de Weryha-Wysoczański (born 1975), art historian, genealogist and writer
 * Jacek Fedorowicz (born 1937), satirist and actor
 * Tova Friedman (born 1938), therapist, social worker, author and Holocaust survivor
 * Eugeniusz Geno Małkowski (1942–2016), painter
 * Gunnar Heinsohn (born 1943), German author, sociologist and economist
 * Klaus Hurrelmann (born 1944), Professor of Public Health and Education
 * Hilary Jastak (1914–2000 in Gdynia), Catholic priest, Doctor of Theology, Chaplain of Solidarity movement, Major of Polish Armed Forces, Lieutenant Commander of the Polish Navy
 * Jan Kaczkowski (1977–2016), Roman Catholic priest, doctor of theological sciences, bioethicist, vlogger, organizer, and director of the Puck Hospice
 * Janusz Kaczmarek (born 1961), lawyer, prosecutor and politician
 * Marcin Kupinski (born 1983), ballet dancer
 * Tomasz Makowiecki (born 1983), musician, singer and songwriter
 * Dorota Nieznalska (born 1973), visual artist and sculptor
 * Kazimierz Ostrowski (1917–1999 in Gdynia), painter
 * Anna Przybylska (1978–2014), actress and model
 * Zvi Aryeh Rosenfeld (1922–1978), Polish-American rabbi and educator
 * Jerzy Rubach (born 1948), Polish and American linguist who specializes in phonology
 * Arkadiusz Rybicki (1953–2010), politician, active in the Solidarity movement
 * Joanna Senyszyn (born 1949), left-wing politician, vice-president of the Democratic Left Alliance (SLD) and MEP
 * Anna Siewierska (1955-2011), Polish-born linguist, specialist in language typology
 * Wojciech Szczurek (born 1963), Mayor of the City of Gdynia since 1998
 * Józef Unrug (1884–1973), German-born Polish vice admiral who helped create the Polish navy
 * Marian Zacharski (born 1951), Intelligence officer convicted of espionage
 * Marek Żukowski (born 1952), theoretical physicist, specializes in quantum mechanics

Sport

 * Teresa Remiszewska (1928–2002), Solo ocean yacht sailor
 * Jörg Berger (1944–2010), German soccer player, trainer
 * Adelajda Mroske (1944–1975), speed skater, she competed in four events at the 1964 Winter Olympics
 * Ryszard Marczak (born 1945), former long-distance runner from Poland, competed in the marathon at the 1980 Summer Olympics
 * Józef Błaszczyk (born 1947), sailor who competed in the 1972 Summer Olympics
 * Andrzej Chudziński (1948–1995), swimmer, competed in three events at the 1972 Summer Olympics
 * Anna Sobczak (born 1967), fencer, competed in the women's individual and team foil events at the 1988 and 1992 Summer Olympics
 * Tomasz Sokołowski (born 1970), footballer, over 350 pro games and 12 for Poland
 * Jarosław Rodzewicz (born 1973), fencer, won a silver medal in the team foil event at the 1996 Summer Olympics
 * Marcin Mięciel (born 1975), soccer player, over 500 pro games
 * Michael Klim (born 1977), Polish-born Australian swimmer, Olympic gold medallist and world champion
 * Anna Rybicka (born 1977), fencer, she won a silver medal in the women's team foil event at the 2000 Summer Olympics
 * Andrzej Bledzewski (born 1977), retired football goalkeeper, over 400 pro games
 * Tomasz Dawidowski (born 1978), footballer, over 200 pro games and 10 for Poland
 * Maciej Grabowski (born 1978), laser class sailor, competed in the 2000, 2004 and 2008 Summer Olympics
 * Adriana Dadci (born 1979), judoka, competed at the 2004 Summer Olympics
 * Stefan Liv (1980–2011), Polish-born Swedish professional ice hockey goaltender
 * Monika Pyrek (born 1980), retired pole vaulter, competed at the 2000, 2004 2008 and 2012 Summer Olympics
 * Anna Rogowska (born 1981), pole vaulter, the bronze medallist at the 2004 Summer Olympics
 * Michał Zych (born 1982), ice dancer
 * Karolina Chlewińska (born 1983), foil fencer, competed at the 2008 Summer Olympics
 * Igor Janik (born 1983), javelin thrower, competed in the 2008 and 2012 Summer Olympics
 * Klaudia Jans-Ignacik (born 1984), retired tennis player, competed in the 2008 and 2012 Summer Olympics
 * Piotr Hallmann (born 1987), mixed martial artist, second lieutenant in the Polish Navy
 * Joanna Mitrosz (born 1988), rhythmic gymnast, competed at the 2008 and 2012 Summer Olympics
 * Małgorzata Białecka (born 1988), windsurfer, competed at 2016 Summer Olympics
 * Olek Czyż (born 1990), professional basketball player, played for Poland
 * Justyna Plutowska (born 1991), ice dancer

Fictional characters

 * Ernst Stavro Blofeld (born 28 May 1908 in Gdingen), fictional character and villain from the James Bond series of novels and films, created by Ian Fleming

Sports
Sport teams
 * Arka Gdynia – men's football team (Polish Cup winner 1979 and 2017, Polish SuperCup winner in 2017 and in 2018. Currently plays in the first division of Polish football, the Ekstraklasa)
 * Bałtyk Gdynia – men's football team, currently playing in Polish 4th division
 * Arka Gdynia (basketball) – men's basketball team (9 time Polish Basketball League winner)
 * Arka Gdynia (women's basketball) – women's basketball team (12-time Basket Liga Kobiet champion)
 * RC Arka Gdynia – rugby team (4-time Polish Champions)
 * Seahawks Gdynia – American football team (Polish American Football League) (4-time champion of Poland in 2012, 2014 and in 2015)
 * Arka Gdynia (handball) – handball team which plays in Ekstraliga (First division of Polish handball)

International events

 * 2017 UEFA European Under-21 Championship
 * 2019 FIFA U-20 World Cup
 * 2020 World Athletics Half Marathon Championships

Port of Gdynia
In 2007, 364,202 passengers, 17,025,000 tons of cargo and containers passed through the port. Regular car ferry service operates between Gdynia and Karlskrona, Sweden.

Public transport
Gdynia operates one of only three trolleybus systems in Poland, alongside Lublin and Tychy. Today there are 18 trolleybus lines in Gdynia with a total length of 96 km. The fleet is modern and consists of Solaris Trollino cars. There is also a historic line, connecting city centre with a district of Orłowo operated by five retro trolleybuses. In addition to that, Gdynia operates an extensive network of bus lines, connecting the city with the adjacent suburbs.

Airport
The conurbation's main airport, Gdańsk Lech Wałęsa Airport, lays approximately 25 km south-west of central Gdynia, and has connections to approximately 55 destinations. It is the third largest airport in Poland. A second General Aviation terminal was scheduled to be opened by May 2012, which will increase the airport's capacity to 5mln passengers per year.

Another local airport, (Gdynia-Kosakowo Airport) is situated partly in the village of Kosakowo, just to the north of the city, and partly in Gdynia. This has been a military airport since the World War II, but it has been decided in 2006 that the airport will be used to serve civilians. Work was well in progress and was due to be ready for 2012 when the project collapsed following a February 2014 EU decision regarding Gdynia city funding as constituting unfair competition to Gdańsk airport. In March 2014, the airport management company filed for bankruptcy, this being formally announced in May that year. The fate of some PLN 100 million of public funds from Gdynia remain unaccounted for with documents not being released, despite repeated requests for such from residents to the city president, Wojciech Szczurek.

Road transport
Trasa Kwiatkowskiego links Port of Gdynia and the city with Obwodnica Trójmiejska, and therefore A1 motorway. National road 6 connects Tricity with Słupsk, Koszalin and Szczecin agglomeration.

Railways
The principal station in Gdynia is Gdynia Główna railway station, the busiest railway station in the Tricity and northern Poland and sixth busiest in Poland overall, serving 13,41 mln passengers in 2022. Gdynia has eleven railway stations. Local train services are provided by the 'Fast Urban Railway,' Szybka Kolej Miejska (Tricity) operating frequent trains covering the Tricity area including Gdańsk, Sopot and Gdynia. Long-distance trains from Warsaw via Gdańsk terminate at Gdynia, and there are direct trains to Szczecin, Poznań, Katowice, Lublin and other major cities. In 2011-2015 the Warsaw-Gdańsk-Gdynia route was undergoing a major upgrading costing $3 billion, partly funded by the European Investment Bank, including track replacement, realignment of curves and relocation of sections of track to allow speeds up to 200 km/h, modernization of stations, and installation of the most modern ETCS signalling system, which was completed in June 2015. In December 2014 new Alstom Pendolino high-speed trains were put into service between Gdynia, Warsaw and Kraków reducing rail travel times to Gdynia by 2 hours.

Economy
Notable companies that have their headquarters or regional offices in Gdynia:
 * PROKOM SA – the largest Polish I.T. company
 * C. Hartwig Gdynia SA – one of the largest Polish freight forwarders
 * Sony Pictures – finance center
 * Thomson Reuters – business data provider
 * Vistal – bridge constructions, offshore and shipbuilding markets; partially located on old Stocznia Gdynia terrains
 * Nauta – ship repair yard; partially located on old Stocznia Gdynia terrains
 * Crist – shipbuilding, offshore constructions, steel structures, sea engineering, civil engineering; located on old Stocznia Gdynia terrains

Former:
 * Stocznia Gdynia – former largest Polish shipyard, now under bankruptcy procedures
 * Nordea – banks, sold and consolidated with PKO bank

Education
There are currently 8 universities and institutions of higher education based in Gdynia. Many students from Gdynia also attend universities located in the Tricity.

 * State-owned:
 * Gdynia Maritime University
 * Polish Naval Academy
 * Privately owned:
 * WSB Merito Universities – WSB Merito University in Gdańsk, departments of Economics and Management
 * Academy of International Economic and Political Relations
 * University of Business and Administration in Gdynia
 * Pomeranian Higher School of Humanities
 * Cardinal Stefan Wyszyński University – department in Gdynia
 * Higher School of Social Communication

Twin towns – sister cities
Gdynia is twinned with:

 * Aalborg, Denmark
 * Baranavichy, Belarus
 * Brooklyn (New York), United States
 * Côte d'Opale (communauté), France
 * Haikou, China
 * Karlskrona, Sweden
 * Kiel, Germany
 * Klaipėda, Lithuania
 * Kotka, Finland
 * Kristiansand, Norway
 * Kunda (Viru-Nigula), Estonia
 * Liepāja, Latvia
 * Plymouth, England, United Kingdom
 * Seattle, United States

Former twin towns:
 * Kaliningrad, Russia (terminated in 2022 due to the Russian invasion of Ukraine)

title: Gdynia


text:
Gluon

A gluon is a type of massless elementary particle that mediates the strong interaction between quarks, acting as the exchange particle for the interaction. Gluons are massless vector bosons, thereby having a spin of 1. Through the strong interaction, gluons bind quarks into groups according to quantum chromodynamics (QCD), forming hadrons such as protons and neutrons.

Gluons carry the color charge of the strong interaction, thereby participating in the strong interaction as well as mediating it. Because gluons carry the color charge, QCD is more difficult to analyze compared to quantum electrodynamics (QED) where the photon carries no electric charge.

The term was coined by Murray Gell-Mann in 1962 for being similar to an adhesive or glue that keeps the nucleus together. Together with the quarks, these particles were referred to as partons by Richard Feynman.

Properties
The gluon is a vector boson, which means it has a spin of 1. While massive spin-1 particles have three polarization states, massless gauge bosons like the gluon have only two polarization states because gauge invariance requires the field polarization to be transverse to the direction that the gluon is traveling. In quantum field theory, unbroken gauge invariance requires that gauge bosons have zero mass. Experiments limit the gluon's rest mass (if any) to less than a few MeV/c2. The gluon has negative intrinsic parity.

Counting gluons
There are eight independent types of gluons in QCD. This is unlike the photon of QED or the three W and Z bosons of the weak interaction.

Additionally, gluons are subject to the color charge phenomena. Quarks carry three types of color charge; antiquarks carry three types of anticolor. Gluons carry both color and anticolor. This gives nine possible combinations of color and anticolor in gluons. The following is a list of those combinations (and their schematic names):
 * red–antired ($r\bar{r}$), red–antigreen ($r\bar{g}$), red–antiblue ($r\bar{b}$)
 * green–antired ($g\bar{r}$), green–antigreen ($g\bar{g}$), green–antiblue ($g\bar{b}$)
 * blue–antired ($b\bar{r}$), blue–antigreen ($b\bar{g}$), blue–antiblue ($b\bar{b}$)

These possible combinations are only effective states, not the actual observed color states of gluons. To understand how they are combined, it is necessary to consider the mathematics of color charge in more detail.

Color singlet states
The stable strongly interacting particles, including hadrons like the proton or the neutron, are observed to be "colorless". More precisely, they are in a "color singlet" state, and mathematically analogous to a spin singlet state. The states allow interaction with other color singlets, but not other color states; because long-range gluon interactions do not exist, this illustrates that gluons in the singlet state do not exist either.

The color singlet state is:
 * $$(r\bar{r}+b\bar{b}+g\bar{g})/\sqrt{3}.$$

If one could measure the color of the state, there would be equal probabilities of it being red–antired, blue–antiblue, or green–antigreen.

Eight color states
There are eight remaining independent color states corresponding to the "eight types" or "eight colors" of gluons. Since the states can be mixed together, there are multiple ways of presenting these states. These are known as the "color octet", and a commonly used list for each is:

These are equivalent to the Gell-Mann matrices. The critical feature of these particular eight states is that they are linearly independent, and also independent of the singlet state, hence 32 − 1 or 23. There is no way to add any combination of these states to produce any others. It is also impossible to add them to make r$\overline{q}$, g$1.3 MeV/c^{2}$, or b$\overline{r}$ the forbidden singlet state. There are many other possible choices, but all are mathematically equivalent, at least equally complicated, and give the same physical results.

Group theory details
Formally, QCD is a gauge theory with SU(3) gauge symmetry. Quarks are introduced as spinors in Nf flavors, each in the fundamental representation (triplet, denoted 3) of the color gauge group, SU(3). The gluons are vectors in the adjoint representation (octets, denoted 8) of color SU(3). For a general gauge group, the number of force-carriers, like photons or gluons, is always equal to the dimension of the adjoint representation. For the simple case of SU(N), the dimension of this representation is N2 − 1.

In group theory, there are no color singlet gluons because quantum chromodynamics has an SU(3) rather than a U(3) symmetry. There is no known a priori reason for one group to be preferred over the other, but as discussed above, the experimental evidence supports SU(3). If the group were U(3), the ninth (colorless singlet) gluon would behave like a "second photon" and not like the other eight gluons.

Confinement
Since gluons themselves carry color charge, they participate in strong interactions. These gluon–gluon interactions constrain color fields to string-like objects called "flux tubes", which exert constant force when stretched. Due to this force, quarks are confined within composite particles called hadrons. This effectively limits the range of the strong interaction to $\overline{g}$ meters, roughly the size of a nucleon. Beyond a certain distance, the energy of the flux tube binding two quarks increases linearly. At a large enough distance, it becomes energetically more favorable to pull a quark–antiquark pair out of the vacuum rather than increase the length of the flux tube.

One consequence of the hadron-confinement property of gluons is that they are not directly involved in the nuclear forces between hadrons. The force mediators for these are other hadrons called mesons.

Although in the normal phase of QCD single gluons may not travel freely, it is predicted that there exist hadrons that are formed entirely of gluons — called glueballs. There are also conjectures about other exotic hadrons in which real gluons (as opposed to virtual ones found in ordinary hadrons) would be primary constituents. Beyond the normal phase of QCD (at extreme temperatures and pressures), quark–gluon plasma forms. In such a plasma there are no hadrons; quarks and gluons become free particles.

Experimental observations
Quarks and gluons (colored) manifest themselves by fragmenting into more quarks and gluons, which in turn hadronize into normal (colorless) particles, correlated in jets. As revealed in 1978 summer conferences, the PLUTO detector at the electron-positron collider DORIS (DESY) produced the first evidence that the hadronic decays of the very narrow resonance Υ(9.46) could be interpreted as three-jet event topologies produced by three gluons. Later, published analyses by the same experiment confirmed this interpretation and also the spin = 1 nature of the gluon (see also the recollection and PLUTO experiments).

In summer 1979, at higher energies at the electron-positron collider PETRA (DESY), again three-jet topologies were observed, now clearly visible and interpreted as q$\overline{b}$ gluon bremsstrahlung, by TASSO, MARK-J and PLUTO experiments (later in 1980 also by JADE ). The spin = 1 property of the gluon was confirmed in 1980 by TASSO and PLUTO experiments (see also the review ). In 1991 a subsequent experiment at the LEP storage ring at CERN again confirmed this result.

The gluons play an important role in the elementary strong interactions between quarks and gluons, described by QCD and studied particularly at the electron-proton collider HERA at DESY. The number and momentum distribution of the gluons in the proton (gluon density) have been measured by two experiments, H1 and ZEUS, in the years 1996–2007. The gluon contribution to the proton spin has been studied by the HERMES experiment at HERA. The gluon density in the proton (when behaving hadronically) also has been measured.

Color confinement is verified by the failure of free quark searches (searches of fractional charges). Quarks are normally produced in pairs (quark + antiquark) to compensate the quantum color and flavor numbers; however at Fermilab single production of top quarks has been shown. No glueball has been demonstrated.

Deconfinement was claimed in 2000 at CERN SPS in heavy-ion collisions, and it implies a new state of matter: quark–gluon plasma, less interactive than in the nucleus, almost as in a liquid. It was found at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven in the years 2004–2010 by four contemporaneous experiments. A quark–gluon plasma state has been confirmed at the CERN Large Hadron Collider (LHC) by the three experiments ALICE, ATLAS and CMS in 2010.

Jefferson Lab's Continuous Electron Beam Accelerator Facility, in Newport News, Virginia, is one of 10 Department of Energy facilities doing research on gluons. The Virginia lab was competing with another facility – Brookhaven National Laboratory, on Long Island, New York – for funds to build a new electron-ion collider. In December 2019, the US Department of Energy selected the Brookhaven National Laboratory to host the electron-ion collider.

External resources

 * Big Think website, clear explanation of the QCD Octet
 * Why are there eight gluons and not nine?

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Gluon


text:
Book of Genesis

The Book of Genesis (from Greek Γένεσις; ; Liber Genesis) is the first book of the Hebrew Bible and the Christian Old Testament. Its Hebrew name is the same as its first word, ('In the beginning'). Genesis is an account of the creation of the world, the early history of humanity, and the origins of the Jewish people.

Genesis is part of the Torah or Pentateuch, the first five books of the Bible. Tradition credits Moses as the Torah's author. It was probably composed around the 5th century BC, although some scholars believe that primeval history (chapters 1–11), may have been composed and added as late as the 3rd century BC. Based on scientific interpretation of archaeological, genetic, and linguistic evidence, most mainstream Bible scholars consider Genesis to be primarily mythological rather than historical.

It is divisible into two parts, the primeval history (chapters 1–11) and the ancestral history (chapters 12–50). The primeval history sets out the author's concepts of the nature of the deity and of humankind's relationship with its maker: God creates a world which is good and fit for humans, but when man corrupts it with sin, God decides to destroy his creation, sparing only the righteous Noah and his family to re-establish the relationship between man and God. The ancestral history (chapters 12–50) tells of the prehistory of Israel, God's chosen people. At God's command, Noah's descendant Abraham journeys from his birthplace (described as Ur of the Chaldeans and whose identification with Sumerian Ur is tentative in modern scholarship) into the God-given land of Canaan, where he dwells as a sojourner, as does his son Isaac and his grandson Jacob. Jacob's name is changed to "Israel", and through the agency of his son Joseph, the children of Israel descend into Egypt, 70 people in all with their households, and God promises them a future of greatness. Genesis ends with Israel in Egypt, ready for the coming of Moses and the Exodus (departure). The narrative is punctuated by a series of covenants with God, successively narrowing in scope from all humankind (the covenant with Noah) to a special relationship with one people alone (Abraham and his descendants through Isaac and Jacob).

In Judaism, the theological importance of Genesis centres on the covenants linking God to his chosen people and the people to the Promised Land. In both Judaism and Christianity, a genre of literature emerged dedicated to interpreting and commenting on the Genesis creation narrative, known as the Hexaemeron.

Title
The name Genesis is from the Latin Vulgate, in turn borrowed or transliterated from Greek Γένεσις, meaning 'origin';, 'In [the] beginning'.

Composition
Genesis was written anonymously, but both Jewish and Christian religious tradition attributes the entire Pentateuch—Genesis, Exodus, Leviticus, Numbers and Deuteronomy—to Moses. During the Enlightenment, the philosophers Benedict Spinoza and Thomas Hobbes questioned Mosaic authorship. In the 17th century, Richard Simon proposed that the Pentateuch was written by multiple authors over a long period of time. The involvement of multiple authors is suggested by internal contradictions within the text. For example, Genesis includes two creation narratives.

By the early 1860s, the leading theory for the Pentateuch's composition was the old supplementary hypothesis. This theory held that the earliest portions, the so-called Book of Origins (containing Genesis 1 and most of the priestly laws in Exodus, Leviticus, and Numbers), was composed in the time of King Solomon by a priest or Levite. This author used the Hebrew word elohim for God. This original work was expanded in the 8th century BC, with the name Yahweh used for God. In the 7th century BC, during the time of Jeremiah, the final parts of the Pentateuch were added, specifically the main parts of Deuteronomy. This would mean the Pentateuch achieved its final form before the Babylonian Exile (c. 598 BC).

At the end of the 19th century, most scholars adopted the documentary hypothesis. This theory held that the five books of the Pentateuch came from four sources: the Yahwist (abbreviated as J), the Elohist (E), the Deuteronomist (D) and the Priestly source (P). Each source was held to tell the same basic story, with the sources later combined by various editors. Scholars were able to distinguish sources based on the designations for God. For example, the Yahwist source uses Yahweh, while the Elohistic and Priestly sources use Elohim. Scholars also use repeated and duplicate stories to identify separate sources. In Genesis, these include the two creation stories, three different wife–sister narratives, and the two versions of Abraham sending Hagar and Ishmael into the desert.

According to the documentary hypothesis, J was produced during the 9th century BC in the southern Kingdom of Judah and was believed to be the earliest source. E was written in the northern Kingdom of Israel during the 8th century BC. D was written in Judah in the 7th century BC and associated with the religious reforms of King Josiah c. 625 BC. The latest source was P, which was written during the 5th century in Babylon. Based on these dates, Genesis and the rest of the Pentateuch did not reach its final, present-day form until after the Babylonian Exile. Julius Wellhausen argued that the Pentateuch was finalized in the time of Ezra. Ezra 7:14 records that Ezra traveled from Babylon to Jerusalem in 458 BC with God's law in his hand. Wellhausen argued that this was the newly compiled Pentateuch. Nehemiah 8–10, according to Wellhausen, describes the publication and public acceptance of this new law code c. 444 BC. There was now a large gap between the earliest sources of the Pentateuch and the period they claimed to describe, which ended c. 1200 BC.

Most scholars held to the documentary hypothesis until the 1980s. Since then, a number of variations and revisions of the documentary hypothesis have been proposed. The new supplementary hypothesis posits three main sources for the Pentateuch: J, D, and P. The E source is considered no more than a variation of J, and P is considered a body of revisions and expansions to the J (or "non-Priestly") material. The Deuteronomistic source does not appear in Genesis. More recent thinking is that J dates from either just before or during the Babylonian Exile, and the Priestly final edition was made late in the Exilic period or soon after. The almost complete absence of all the characters and incidents mentioned in primeval history from the rest of the Hebrew Bible has led a sizeable minority of scholars to conclude that these chapters were composed much later than those that follow, possibly in the 3rd century BC.

As for why the book was created, a theory which has gained considerable interest, although still controversial, is that of Persian imperial authorisation. This proposes that the Persians of the Achaemenid Empire, after their conquest of Babylon in 539 BC, agreed to grant Jerusalem a large measure of local autonomy within the empire, but required the local authorities to produce a single law code accepted by the entire community. The two powerful groups making up the community—the priestly families who controlled the Second Temple and who traced their origin to Moses and the wilderness wanderings, and the major landowning families who made up the "elders" and who traced their own origins to Abraham, who had "given" them the land—were in conflict over many issues, and each had its own "history of origins". However, the Persian promise of greatly increased local autonomy for all provided a powerful incentive to cooperate in producing a single text.

Genre
Genesis is an example of a work in the "antiquities" genre, as the Romans knew it, a popular genre telling of the appearance of humans and their ancestors and heroes, with elaborate genealogies and chronologies fleshed out with stories and anecdotes. Notable examples are found in the work of Greek historians of the 6th century BC: their intention was to connect notable families of their own day to a distant and heroic past, and in doing so they did not distinguish between myth, legend, and facts. Professor Jean-Louis Ska of the Pontifical Biblical Institute calls the basic rule of the antiquarian historian the "law of conservation": everything old is valuable, nothing is eliminated. This antiquity was needed to prove the worth of Israel's traditions to the nations (the neighbours of the Jews in the early Persian province of Judea), and to reconcile and unite the various factions within Israel itself.

Describing the work of the biblical authors, John Van Seters wrote that lacking many historical traditions and none from the distant past, "They had to use myths and legends for earlier periods. In order to make sense out of the variety of different and often conflicting versions of stories, and to relate the stories to each other, they fitted them into a genealogical chronology." Tremper Longman describes Genesis as theological history: "the fact that these events took place is assumed, and not argued. The concern of the text is not to prove the history but rather to impress the reader with the theological significance of these acts".

Textual witnesses
There are four major textual witnesses to the book: the Masoretic Text, the Samaritan Pentateuch, the Septuagint, and fragments of Genesis found at Qumran. The Qumran group provides the oldest manuscripts but covers only a small proportion of the book; in general, the Masoretic Text is well preserved and reliable, but there are many individual instances where the other versions preserve a superior reading.

Structure
Genesis appears to be structured around the recurring phrase, meaning "these are the generations", with the first use of the phrase referring to the "generations of heaven and earth" and the remainder marking individuals. The formula, occurring eleven times in the book of Genesis, serves as a heading which marks a transition to a new subject. The creation account of Genesis 1 functions as a prologue for the whole book and is not introduced with a. The divide the book into the following sections:

 * 1) Genesis 1:1–2:3 In the beginning (prologue)
 * 2) Genesis 2:4–4:26  of Heaven and Earth (narrative)
 * 3) Genesis 5:1–6:8  of Adam (genealogy, )
 * 4) Genesis 6:9–9:29  of Noah (Genesis flood narrative)
 * 5) Genesis 10:1–11:9  of Noah's sons Shem, Ham, and Japheth (genealogy)
 * 6) Genesis 11:10–26  of Shem (genealogy)
 * 7) Genesis 11:27–25:11  of Terah (Abraham narrative)
 * 8) Genesis 25:12–18  of Ishmael (genealogy)
 * 9) Genesis 25:19–35:29  of Isaac (Jacob narrative)
 * 10) Genesis 36:1–36:8  of Esau (genealogy)
 * 11) Genesis 36:9–37:1  of Esau "the father of the Edomites" (genealogy)
 * 12) Genesis 37:2–50:26  of Jacob (Joseph narrative)

It is not clear, however, what this meant to the original authors, and most modern commentators divide it into two parts based on the subject matter, a primeval history (chapters 1–11) and a patriarchal history (chapters 12–50). While the first is far shorter than the second, it sets out the basic themes and provides an interpretive key for understanding the entire book. The primeval history has a symmetrical structure hinging on the flood story (chapters 6–9) with the events before the flood mirrored by the events after. The ancestral history is structured around the three patriarchs Abraham, Jacob and Joseph. The stories of Isaac arguably do not make up a coherent cycle of stories and function as a bridge between the cycles of Abraham and Jacob.

Primeval history (chapters 1–11)

The Genesis creation narrative comprises two different stories; the first two chapters roughly correspond to these. In the first, Elohim, the generic Hebrew word for God, creates the heavens and the earth including humankind, in six days, and rests on the seventh. In the second, God, now referred to as "Yahweh Elohim" (rendered as "the God" in English translations), creates two individuals, Adam and Eve, as the first man and woman, and places them in the Garden of Eden.

In the third chapter, God instructs them not to eat the fruit of the tree of the knowledge of good and evil. They promise not to, but a talking serpent, portrayed as a deceptive creature or trickster, convinces Eve to eat the fruit against God's wishes, and she convinces Adam, whereupon God throws them out and curses both of them—Adam was cursed with getting what he needs only by sweat and work, and Eve to giving birth in pain. This is interpreted by Christians as the "fall of man" into sin. Eve bears two sons, Cain and Abel. Cain works in the garden, and Abel works with meat; they both offer offerings to God one day, and God does not accept Cain's offering but does accept Abel's. This causes Cain to resent Abel, and Cain ends up murdering him. God then curses Cain. Eve bears another son, Seth, to take Abel's place in accordance to the promises given at 3:15, 20.

After many generations of Adam have passed from the lines of Cain and Seth, the world becomes corrupted by human sin and Nephilim, and God wants to wipe out humanity for their wickedness. However, Noah is the only good human; so first, he instructs the righteous Noah and his family to build an ark and put examples of all the animals on it, seven pairs of every clean animal and one pair of every unclean. Then God sends a great flood to wipe out the rest of the world. When the waters recede, God promises he will never destroy the world with water again, making a rainbow as a symbol of his promise. God sees humankind cooperating to build a great tower city, the Tower of Babel, and divides humanity with many languages and sets them apart with confusion. Then, a generation line from Shem to Abram is described.

Patriarchal age (chapters 12–50)

Abram, a man descended from Noah, is instructed by God to travel from his home in Mesopotamia to the land of Canaan. There, God makes a promise to Abram, promising that his descendants shall be as numerous as the stars, but that people will suffer oppression in a foreign land for four hundred years, after which they will inherit the land "from the river of Egypt to the great river, the river Euphrates". Abram's name is changed to 'Abraham' and that of his wife Sarai to Sarah (meaning 'princess'), and God says that all males should be circumcised as a sign of his promise to Abraham. Due to her old age, Sarah tells Abraham to take her Egyptian handmaiden, Hagar, as a second wife (to bear a child). Through Hagar, Abraham fathers Ishmael.

God then plans to destroy the cities of Sodom and Gomorrah for the sins of their people. Abraham protests but fails to get God to agree not to destroy the cities (reasoning with Abraham that not even ten righteous persons were found there; and among the righteous was Abraham's nephew Lot). Angels save Abraham's nephew Lot (who was living there at the same time) and his family, but his wife looks back on the destruction, (even though God commanded not to) and turns into a pillar of salt for going against his word. Lot's daughters, concerned that they are fugitives who will never find husbands, get Lot drunk so they can become pregnant by him, and give birth to the ancestors of the Moabites and Ammonites.

Abraham and Sarah go to the Philistine town of Gerar, pretending to be brother and sister (they are half-siblings). The King of Gerar takes Sarah for his wife, but God warns him to return her (as she is really Abraham's wife) and he obeys. God sends Sarah a son and tells her she should name him Isaac; through him will be the establishment of the covenant (promise). Sarah then drives Ishmael and his mother Hagar out into the wilderness (because Ishmael is not her real son and Hagar is a slave), but God saves them and promises to make Ishmael a great nation.

Then, God tests Abraham by demanding that he sacrifice Isaac. As Abraham is about to lay the knife upon his son, "the Angel of the Lord" restrains him, promising him again innumerable descendants. On the death of Sarah, Abraham purchases Machpelah (believed to be modern Hebron) for a family tomb and sends his servant to Mesopotamia to find among his relations a wife for Isaac; after proving herself worthy, Rebekah becomes Isaac's betrothed. Keturah, Abraham's other wife, births more children, among whose descendants are the Midianites. Abraham dies at a prosperous old age and his family lays him to rest in Hebron (Machpelah).

Isaac's wife Rebekah gives birth to the twins Esau (meaning 'velvet'), father of the Edomites, and Jacob (meaning 'supplanter' or 'follower'). Esau was a couple of seconds older as he had come out of the womb first, and was going to become the heir; however, through carelessness, he sold his birthright to Jacob for a bowl of stew. His mother, Rebekah, ensures Jacob rightly gains his father's blessing as the firstborn son and inheritor. At 77 years of age, Jacob leaves his parents and later seeks a wife and meets Rachel at a well. He goes to her father, his uncle, where he works for a total of 14 years to earn his wives, Rachel and Leah. Jacob's name is changed to Israel after his wrestle with an angel, and by his wives and their handmaidens he has twelve sons, the ancestors of the twelve tribes of the children of Israel, and a daughter, Dinah.

Shechem, son of Hamor the Hivite, rapes Dinah and asks his father to get Dinah for him as his wife, according to Chapter 34. Jacob agrees to the marriage but requires that all the males of Hamor's tribe be circumcised, including Hamor and Shechem. After this was performed and all the men were still weak, Jacob's sons Simeon and Levi murdered all the males. Jacob complained that their act would mean retribution by others, namely the Canaanites and Perizzites. Jacob and his tribe took all the Hivite women and children as well as livestock and other property for themselves.

Joseph, Jacob's favourite son of the twelve, makes his brothers jealous (especially because of special gifts Jacob gave him) and because of that jealousy they sell Joseph into slavery in Egypt. Joseph endures many trials including being innocently sentenced to jail but he stays faithful to God. After several years, he prospers there after the pharaoh of Egypt asks him to interpret a dream he had about an upcoming famine, which Joseph does through God. He is then made second in command of Egypt by the grateful pharaoh, and later on, he is reunited with his father and brothers, who fail to recognize him and plead for food as the famine had reached Canaan as well. After much manipulation to see if they still hate him, Joseph reveals himself, forgives them for their actions, and lets them and their households into Egypt, where Pharaoh assigns to them the land of Goshen. Jacob calls his sons to his bedside and reveals their future before he dies. Joseph lives to old age and tells his brothers before his death that if God leads them out of the country, then they should take his bones with them.

Promises to the ancestors
In 1978, David Clines published The Theme of the Pentateuch. Considered influential as one of the first authors to take up the question of the overarching theme of the Pentateuch, Clines' conclusion was that the overall theme is "the partial fulfilment—which implies also the partial nonfulfillment—of the promise to or blessing of the Patriarchs". (By calling the fulfilment "partial", Clines was drawing attention to the fact that at the end of Deuteronomy the people of Israel are still outside Canaan.)

The patriarchs, or ancestors, are Abraham, Isaac and Jacob, with their wives (Joseph is normally excluded). Since the name YHWH had not been revealed to them, they worshipped El in his various manifestations. (It is, however, worth noting that in the Jahwist source, the patriarchs refer to deity by the name YHWH, for example in Genesis 15.) Through the patriarchs, God announces the election of Israel, that is, he chooses Israel to be his special people and commits himself to their future. God tells the patriarchs that he will be faithful to their descendants (i.e. to Israel), and Israel is expected to have faith in God and his promise. ("Faith" in the context of Genesis and the Hebrew Bible means an agreement to the promissory relationship, not a body of a belief.)

The promise itself has three parts: offspring, blessings, and land. The fulfilment of the promise to each patriarch depends on having a male heir, and the story is constantly complicated by the fact that each prospective mother—Sarah, Rebekah and Rachel—is barren. The ancestors, however, retain their faith in God and God in each case gives a son—in Jacob's case, twelve sons, the foundation of the chosen Israelites. Each succeeding generation of the three promises attains a more rich fulfilment, until through Joseph "all the world" attains salvation from famine, and by bringing the children of Israel down to Egypt he becomes the means through which the promise can be fulfilled.

God's chosen people
Scholars generally agree that the theme of divine promise unites the patriarchal cycles, but many would dispute the efficacy of trying to examine Genesis' theology by pursuing a single overarching theme, instead citing as more productive the analysis of the Abraham cycle, the Jacob cycle, and the Joseph cycle, and the Yahwist and Priestly sources. The problem lies in finding a way to unite the patriarchal theme of the divine promise to the stories of Genesis 1–11 (the primeval history) with their theme of God's forgiveness in the face of man's evil nature. One solution is to see the patriarchal stories as resulting from God's decision not to remain alienated from humankind: God creates the world and humans, humans rebel, and God "elects" (chooses) Abraham.

To this basic plot (which comes from the Yahwist), the Priestly source has added a series of covenants dividing history into stages, each with its own distinctive "sign". The first covenant is between God and all living creatures, and is marked by the sign of the rainbow; the second is with the descendants of Abraham (Ishmaelites and others as well as Israelites), and its sign is circumcision; and the last, which does not appear until the Book of Exodus, is with Israel alone, and its sign is Sabbath. A great leader mediates each covenant (Noah, Abraham, Moses), and at each stage God progressively reveals himself by his name (Elohim with Noah, El Shaddai with Abraham, Yahweh with Moses).

Deception
Throughout Genesis, various figures engage in deception or trickery to survive or prosper. Biblical scholar David M. Carr notes that such stories reflect the vulnerability felt by ancient Israelites and that "such stories can be a major way of gaining hope and resisting domination". Examples include:
 * To avoid being killed, a patriarch (Abraham in 12:10–20 and 20:1–18 and Isaac in 26:6–11) tells a king that his wife is actually his sister.
 * In chapter 25, Jacob tricks Esau into selling his birthright for a pot of lentil stew.
 * In chapter 27, Isaac is tricked by Rebekah into giving Jacob the superior blessing instead of Esau.
 * In chapter 29, Jacob believes he is marrying Rachel but is tricked into marrying her sister.

Cultural impact
By totaling the spans of time in the genealogies of Genesis, religious authorities have calculated what they consider to be the age of the world since creation. This Anno Mundi system of counting years is the basis of the Hebrew calendar and Byzantine calendar. Counts differ somewhat, but they generally place the age of the Earth at about six thousand years.

During the Protestant Reformation, rivalry between Catholic and Protestant Christians led to a closer study of the Bible and a competition to take its words more seriously. Thus, scholars in Europe from the 16th to the 19th century treated the book of Genesis as factual. As evidence in the fields of paleontology, geology and other sciences was uncovered, scholars tried to fit these discoveries into the Genesis creation account. For example, Johann Jakob Scheuchzer in the 18th century believed that fossils were the remains of creatures killed during the flood. This literal understanding of Genesis fell out of favor with scholars during the Victorian crisis of faith as evidence mounted that the Earth was far older than six thousand years.

Judaism's weekly Torah portions
It is a custom among religious Jewish communities for a weekly Torah portion, popularly referred to as a, to be read during Jewish prayer services on Saturdays, Mondays and Thursdays. The full name, פָּרָשַׁת הַשָּׁבוּעַ, is popularly abbreviated to (also   or ), and is also known as a  (or  ).

The is a section of the Torah (Five Books of Moses) used in Jewish liturgy during a particular week. There are 54 weekly parshas, or in Hebrew, and the full cycle is read over the course of one Jewish year.

The first 12 of the 54 come from the Book of Genesis, and they are:
 * 1) Chapters 1–6 (verses 1–8) Parashat Bereshit
 * 2) Chapters 6 (v. 9 ff)–11 Parashat Noach
 * 3) Chapters 12–17 Parashat Lekh Lekha
 * 4) Chapters 18–22 Parashat Vayera
 * 5) Chapters 23–25 (v. 1–18) Parashat Chayyei Sarah
 * 6) Chapters 25 (v. 19 ff)–28 (v. 1–9) Parashat Toledot
 * 7) Chapters 28 (v. 10 ff)–32 (v. 1–3) Parashat Vayetzei
 * 8) Chapters 32 (v. 4 ff)–36 Parashat Vayishlach
 * 9) Chapters 37–40 Parashat Vayeshev
 * 10) Chapters 41–44 (v. 1–17) Parashat Miketz
 * 11) Chapters 44 (v. 18 ff)–47 (v. 1–27) Parashat Vayigash
 * 12) Chapters 47 (v. 28 ff)–50 Parashat Vayechi

Commentaries

 * Fretheim, Terence E. "The Book of Genesis." In The New Interpreter's Bible. Edited by Leander E. Keck, vol. 1, pp. 319–674. Nashville: Abingdon Press, 1994. ISBN 0-687-27814-7.
 * Hirsch, Samson Raphael. The Pentateuch: Genesis. Translated by Isaac Levy. Judaica Press, 2nd edition 1999. ISBN 0-910818-12-6. Originally published as Der Pentateuch uebersetzt und erklaert Frankfurt, 1867–1878.
 * Kass, Leon R. The Beginning of Wisdom: Reading Genesis. New York: Free Press, 2003. ISBN 0-7432-4299-8.
 * Plaut, Gunther. The Torah: A Modern Commentary (1981), ISBN 0-8074-0055-6
 * Sarna, Nahum M. The JPS Torah Commentary: Genesis: The Traditional Hebrew Text with the New JPS Translation. Philadelphia: Jewish Publication Society, 1989. ISBN 0-8276-0326-6.
 * Speiser, E.A. Genesis: Introduction, Translation, and Notes. New York: Anchor Bible, 1964. ISBN 0-385-00854-6.
 * Hirsch, Samson Raphael. The Pentateuch: Genesis. Translated by Isaac Levy. Judaica Press, 2nd edition 1999. ISBN 0-910818-12-6. Originally published as Der Pentateuch uebersetzt und erklaert Frankfurt, 1867–1878.
 * Kass, Leon R. The Beginning of Wisdom: Reading Genesis. New York: Free Press, 2003. ISBN 0-7432-4299-8.
 * Plaut, Gunther. The Torah: A Modern Commentary (1981), ISBN 0-8074-0055-6
 * Sarna, Nahum M. The JPS Torah Commentary: Genesis: The Traditional Hebrew Text with the New JPS Translation. Philadelphia: Jewish Publication Society, 1989. ISBN 0-8276-0326-6.
 * Speiser, E.A. Genesis: Introduction, Translation, and Notes. New York: Anchor Bible, 1964. ISBN 0-385-00854-6.
 * Sarna, Nahum M. The JPS Torah Commentary: Genesis: The Traditional Hebrew Text with the New JPS Translation. Philadelphia: Jewish Publication Society, 1989. ISBN 0-8276-0326-6.
 * Speiser, E.A. Genesis: Introduction, Translation, and Notes. New York: Anchor Bible, 1964. ISBN 0-385-00854-6.
 * Speiser, E.A. Genesis: Introduction, Translation, and Notes. New York: Anchor Bible, 1964. ISBN 0-385-00854-6.

title: Book of Genesis


text:
Talk:Book of Genesis

Revert
I have reverted an edit which is thoroughly WP:FRINGE, i.e. pseudohistory. tgeorgescu (talk) 13:43, 9 March 2024 (UTC)

 * Yeah, that's totally fine. Thanks for keeping up with these edits and reverting when necessary. Hist9600 (talk) 14:24, 9 March 2024 (UTC)
 * Thank you. Another IP, possibly the same person, had added the same information yesterday, and I had to correct it. It seems someone may be actively watching the page to insert these dates. Ltwin (talk) 16:33, 9 March 2024 (UTC)
 * Agree, both geolocate to Adelaide, Australia. tgeorgescu (talk) 16:55, 9 March 2024 (UTC)

Dead Sea study and date of composition
I just noticed and removed the following from the Composition section:

After checking the article out, I believe this is a misreading of the article. The study authors were attempting to use biblical evidence to determine historical fluctuations in the Dead Sea levels. I don't think they are making definitive statements about when biblical books were actually composed. I could be wrong, however, so I'm leaving this here for other editors to evaluate. Ltwin (talk) 21:51, 9 March 2024 (UTC)

Wiki Education assignment: Criticism as Praxis
— Assignment last updated by Keegancroteau12 (talk) 04:30, 24 April 2024 (UTC)

Does the "myth" description need more subtlety?
In saying "most mainstream Bible scholars consider Genesis to be primarily mythological rather than historical" does this require more subtlety? Sure, Gen.1-11 is mythological, as discussed well in its "primeval history" article. That's basically fine. But Gen.12-50 has very different genre characteristics; indeed the word "myth" never appears in the main body of its "ancestral history" article.

So isn't the statement about the whole book "...consider Genesis to be primarily mythological rather than historical" too blunt in not distinguishing the different genres across the primeval/ancestral divide?

Note: I'm basing this on the summary in the article lead. But the myth-description in the main body (section "Genre") has the same over-generalisation problem, so it's probably better to start from there, then to reflect an appropriate summary back onto the lead.

Feline Hymnic (talk) 16:45, 9 May 2024 (UTC)

 * Subtlety is nice, but I believe the passage in question adequately characterizes the whole text, as the purpose throughout both the primeval and ancestral history halves is to explain and interconnect the origins of the world and the Jewish people within it. Remsense  诉  16:49, 9 May 2024 (UTC)

title: Talk:Book of Genesis


text:
GM

GM or Gm may refer to:

Companies

 * General Motors, US automobile manufacturing company which was founded in 1908
 * Motors Liquidation Company, the US automobile manufacturing company known as General Motors Corporation from 1916 to 2009
 * General Mills, US food manufacturing company
 * Gunn & Moore, UK sports equipment company

Places

 * The Gambia, by ISO 3166 code
 * Germany, by FIPS 10-4 country code
 * Greater Manchester, England

Sports and gaming

 * RGM-79 GM, a mobile suit series in the video game Mobile Suit Gundam
 * Gamemaster or game master, a person officiating in a multiplayer role-playing game
 * Grandmaster (chess)
 * Grandmaster (martial arts)
 * GM (magazine)

Science and measurement

 * Geiger–Müller tube or G-M tube, a type of radiation detector
 * Genetic modification or genetically modified, manipulation of an organism's genome
 * Gigametre or gigameter (Gm), one billion metres
 * Silty gravel, in the Unified Soil Classification System
 * Standard gravitational parameter $$\mu=GM \ $$
 * Metacentric height, related to ship stability
 * Transconductance, an electronic component parameter
 * GM, unit of cross section named after Maria Goeppert Mayer

Other uses

 * Gm (digraph)
 * GM chassis or Gusenichnaya Machina, a Russian tracked vehicle chassis
 * G minor, chords and scale
 * General manager
 * General MIDI, a standardized specification for music synthesizers
 * Genital mutilation
 * George Medal, a UK, primarily civil, decoration for acts of great bravery
 * Gold master, a version of software to be distributed to customers, i.e. ready for release to manufacturing
 * Google Maps
 * Gunner's mate, a US Navy rating

title: GM


text:
Galois group

In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the polynomials that give rise to them via Galois groups is called Galois theory, so named in honor of Évariste Galois who first discovered them.

For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory.

Definition
Suppose that $$E$$ is an extension of the field $$F$$ (written as $$E/F$$ and read "E over F). An automorphism of $$E/F$$ is defined to be an automorphism of $$E$$ that fixes $$F$$ pointwise. In other words, an automorphism of $$E/F$$ is an isomorphism $$\alpha:E\to E$$ such that $$\alpha(x) = x$$ for each $$x\in F$$. The set of all automorphisms of $$E/F$$ forms a group with the operation of function composition. This group is sometimes denoted by $$\operatorname{Aut}(E/F).$$

If $$E/F$$ is a Galois extension, then $$\operatorname{Aut}(E/F)$$ is called the Galois group of $$E/F$$, and is usually denoted by $$\operatorname{Gal}(E/F)$$.

If $$E/F$$ is not a Galois extension, then the Galois group of $$E/F$$ is sometimes defined as $$\operatorname{Aut}(K/F)$$, where $$K$$ is the Galois closure of $$E$$.

Galois group of a polynomial
Another definition of the Galois group comes from the Galois group of a polynomial $$f \in F[x]$$. If there is a field $$K/F$$ such that $$f$$ factors as a product of linear polynomials

 * $$f(x) = (x-\alpha_1)\cdots (x - \alpha_k) \in K[x]$$

over the field $$K$$, then the Galois group of the polynomial $$f$$ is defined as the Galois group of $$K/F$$ where $$K$$ is minimal among all such fields.

Fundamental theorem of Galois theory
One of the important structure theorems from Galois theory comes from the fundamental theorem of Galois theory. This states that given a finite Galois extension $$K/k$$, there is a bijection between the set of subfields $$k \subset E \subset K$$ and the subgroups $$H \subset G.$$ Then, $$E$$ is given by the set of invariants of $$K$$ under the action of $$H$$, so

 * $$E = K^H = \{ a\in K : ga = a \text{ where } g \in H \}$$

Moreover, if $$H$$ is a normal subgroup then $$G/H \cong \operatorname{Gal}(E/k)$$. And conversely, if $$E/k$$ is a normal field extension, then the associated subgroup in $$\operatorname{Gal}(K/k)$$ is a normal group.

Lattice structure
Suppose $$K_1,K_2$$ are Galois extensions of $$k$$ with Galois groups $$G_1,G_2.$$ The field $$K_1K_2$$ with Galois group $$G = \operatorname{Gal}(K_1K_2/k)$$ has an injection $$G \to G_1 \times G_2$$ which is an isomorphism whenever $$K_1 \cap K_2 = k$$.

Inducting
As a corollary, this can be inducted finitely many times. Given Galois extensions $$K_1,\ldots, K_n / k$$ where $$K_{i+1} \cap (K_1\cdots K_i) = k,$$ then there is an isomorphism of the corresponding Galois groups:

 * $$\operatorname{Gal}(K_1\cdots K_n/k) \cong \operatorname{Gal}(K_1/k)\times \cdots \times \operatorname{Gal}(K_n/k).$$

Examples
In the following examples $$F$$ is a field, and $$\Complex, \R, \Q$$ are the fields of complex, real, and rational numbers, respectively. The notation $F(a)$ indicates the field extension obtained by adjoining an element $a$ to the field $F$.

Cardinality of the Galois group and the degree of the field extension
One of the basic propositions required for completely determining the Galois groups of a finite field extension is the following: Given a polynomial $$f(x) \in F[x]$$, let $$E/F$$ be its splitting field extension. Then the order of the Galois group is equal to the degree of the field extension; that is,

 * $$\left|\operatorname{Gal}(E/F)\right| = [E:F]$$

Eisenstein's criterion
A useful tool for determining the Galois group of a polynomial comes from Eisenstein's criterion. If a polynomial $$f \in F[x]$$ factors into irreducible polynomials $$f = f_1\cdots f_k$$ the Galois group of $$f$$ can be determined using the Galois groups of each $$f_i$$ since the Galois group of $$f$$ contains each of the Galois groups of the $$f_i.$$

Trivial group
$$\operatorname{Gal}(F/F)$$ is the trivial group that has a single element, namely the identity automorphism.

Another example of a Galois group which is trivial is $$\operatorname{Aut}(\R/\Q).$$ Indeed, it can be shown that any automorphism of $$\R$$ must preserve the ordering of the real numbers and hence must be the identity.

Consider the field $$K = \Q(\sqrt[3]{2}).$$ The group $$\operatorname{Aut}(K/\Q)$$ contains only the identity automorphism. This is because $$K$$ is not a normal extension, since the other two cube roots of $$2$$,

 * $$\exp \left (\tfrac{2\pi i}{3} \right ) \sqrt[3]{2}$$ and $$\exp \left (\tfrac{4\pi i}{3} \right ) \sqrt[3]{2},$$

are missing from the extension—in other words $K$ is not a splitting field.

Finite abelian groups
The Galois group $$\operatorname{Gal}(\Complex/\R)$$ has two elements, the identity automorphism and the complex conjugation automorphism.

Quadratic extensions
The degree two field extension $$\Q(\sqrt{2})/\Q$$ has the Galois group $$\operatorname{Gal}(\Q(\sqrt{2})/\Q)$$ with two elements, the identity automorphism and the automorphism $$\sigma$$ which exchanges $$\sqrt2$$ and $$-\sqrt2$$. This example generalizes for a prime number $$p \in \N.$$

Product of quadratic extensions
Using the lattice structure of Galois groups, for non-equal prime numbers $$p_1, \ldots, p_k$$ the Galois group of $$\Q \left (\sqrt{p_1},\ldots, \sqrt{p_k} \right)/\Q$$ is

 * $$\operatorname{Gal} \left (\Q(\sqrt{p_1},\ldots, \sqrt{p_k})/\Q \right ) \cong \operatorname{Gal}\left (\Q(\sqrt{p_1})/\Q \right )\times \cdots \times \operatorname{Gal} \left (\Q(\sqrt{p_k})/\Q \right ) \cong (\Z/2\Z)^k$$

Cyclotomic extensions
Another useful class of examples comes from the splitting fields of cyclotomic polynomials. These are polynomials $$\Phi_n$$ defined as

 * $$\Phi_n(x) = \prod_{\begin{matrix} 1 \leq k \leq n \\ \gcd(k,n) = 1\end{matrix}} \left(x-e^{\frac{2ik\pi}{n}} \right)$$

whose degree is $$\phi(n)$$, Euler's totient function at $$n$$. Then, the splitting field over $$\Q$$ is $$\Q(\zeta_n)$$ and has automorphisms $$\sigma_a$$ sending $$\zeta_n \mapsto \zeta_n^a$$ for $$1 \leq a < n$$ relatively prime to $$n$$. Since the degree of the field is equal to the degree of the polynomial, these automorphisms generate the Galois group. If $$n = p_1^{a_1}\cdots p_k^{a_k},$$ then

 * $$\operatorname{Gal}(\Q(\zeta_n)/\Q) \cong \prod_{a_i} \operatorname{Gal}\left (\Q(\zeta_{p_i^{a_i}})/\Q \right )$$

If $$n$$ is a prime $$p $$, then a corollary of this is

 * $$\operatorname{Gal}(\Q(\zeta_p)/\Q) \cong \Z/(p-1)\Z$$

In fact, any finite abelian group can be found as the Galois group of some subfield of a cyclotomic field extension by the Kronecker–Weber theorem.

Finite fields
Another useful class of examples of Galois groups with finite abelian groups comes from finite fields. If $q$ is a prime power, and if $$F = \mathbb{F}_q$$ and $$E=\mathbb{F}_{q^n}$$ denote the Galois fields of order $$q$$ and $$q^n$$ respectively, then $$\operatorname{Gal}(E/F)$$ is cyclic of order $n$ and generated by the Frobenius homomorphism.

Degree 4 examples
The field extension $$\Q(\sqrt{2},\sqrt{3})/\Q$$ is an example of a degree $$4$$ field extension. This has two automorphisms $$\sigma, \tau$$ where $$\sigma(\sqrt{2}) = -\sqrt{2}$$ and $$\tau(\sqrt{3})=-\sqrt{3}.$$ Since these two generators define a group of order $$4$$, the Klein four-group, they determine the entire Galois group.

Another example is given from the splitting field $$E/\Q$$ of the polynomial

 * $$f(x) = x^4 + x^3 + x^2 + x + 1$$

Note because $$(x-1)f(x)= x^5-1,$$ the roots of $$f(x)$$ are $$\exp \left (\tfrac{2k\pi i}{5} \right).$$ There are automorphisms

 * $$\begin{cases}\sigma_l : E \to E \\ \exp \left (\frac{2\pi i}{5} \right) \mapsto \left (\exp \left (\frac{2\pi i}{5} \right ) \right )^l \end{cases}$$

generating a group of order $$4$$. Since $$\sigma_2$$ generates this group, the Galois group is isomorphic to $$\Z/4\Z$$.

Finite non-abelian groups
Consider now $$L = \Q(\sqrt[3]{2}, \omega),$$ where $$\omega$$ is a primitive cube root of unity. The group $$\operatorname{Gal}(L/\Q)$$ is isomorphic to $S_{3}$, the dihedral group of order 6, and $L$ is in fact the splitting field of $$x^3-2$$ over $$\Q.$$

Quaternion group
The Quaternion group can be found as the Galois group of a field extension of $$\Q$$. For example, the field extension

 * $$\Q \left (\sqrt{2}, \sqrt{3}, \sqrt{(2+\sqrt{2})(3+\sqrt{3})} \right )$$

has the prescribed Galois group.

Symmetric group of prime order
If $$f$$ is an irreducible polynomial of prime degree $$p$$ with rational coefficients and exactly two non-real roots, then the Galois group of $$f$$ is the full symmetric group $$S_p.$$

For example, $$f(x)=x^5-4x+2 \in \Q[x]$$ is irreducible from Eisenstein's criterion. Plotting the graph of $$f$$ with graphing software or paper shows it has three real roots, hence two complex roots, showing its Galois group is $$S_5$$.

Comparing Galois groups of field extensions of global fields
Given a global field extension $$K/k$$ (such as $$\mathbb{Q}(\sqrt[5]{3},\zeta_5 )/\mathbb{Q}$$) and equivalence classes of valuations $$w$$ on $$K$$ (such as the $$p$$-adic valuation) and $$v$$ on $$k$$ such that their completions give a Galois field extension"$K_w/k_v$"of local fields, there is an induced action of the Galois group $$G = \operatorname{Gal}(K/k)$$ on the set of equivalence classes of valuations such that the completions of the fields are compatible. This means if $$s \in G$$ then there is an induced isomorphism of local fields"$s_w:K_w \to K_{sw}$"Since we have taken the hypothesis that $$w$$ lies over $$v$$ (i.e. there is a Galois field extension $$K_w/k_v$$), the field morphism $$s_w$$ is in fact an isomorphism of $$k_v$$-algebras. If we take the isotropy subgroup of $$G$$ for the valuation class $$w$$"$G_w = \{s \in G : sw = w \}$"then there is a surjection of the global Galois group to the local Galois group such that there is an isomorphism between the local Galois group and the isotropy subgroup. Diagrammatically, this means $$\begin{matrix} \operatorname{Gal}(K/v)& \twoheadrightarrow & \operatorname{Gal}(K_w/k_v) \\ \downarrow & & \downarrow \\ G & \twoheadrightarrow & G_w \end{matrix}$$ where the vertical arrows are isomorphisms. This gives a technique for constructing Galois groups of local fields using global Galois groups.

Infinite groups
A basic example of a field extension with an infinite group of automorphisms is $$\operatorname{Aut}(\Complex/\Q)$$, since it contains every algebraic field extension $$E/\Q$$. For example, the field extensions $$\Q(\sqrt{a})/\Q$$ for a square-free element $$a \in \Q$$ each have a unique degree $$2$$ automorphism, inducing an automorphism in $$\operatorname{Aut}(\Complex/\Q).$$

One of the most studied classes of infinite Galois group is the absolute Galois group, which is an infinite, profinite group defined as the inverse limit of all finite Galois extensions $$E/F$$ for a fixed field. The inverse limit is denoted

 * $$\operatorname{Gal}(\overline{F}/F) := \varprojlim_{E/F \text{ finite separable}}{\operatorname{Gal}(E/F)}$$,

where $$\overline{F}$$ is the separable closure of the field $$F$$. Note this group is a topological group. Some basic examples include $$\operatorname{Gal}(\overline{\Q}/\Q)$$ and

 * $$\operatorname{Gal}(\overline{\mathbb{F}}_q/\mathbb{F}_q) \cong \hat{\Z} \cong \prod_p \Z_p$$.

Another readily computable example comes from the field extension $$\Q(\sqrt{2},\sqrt{3},\sqrt{5}, \ldots)/ \Q$$ containing the square root of every positive prime. It has Galois group

 * $$\operatorname{Gal}(\Q(\sqrt{2},\sqrt{3},\sqrt{5}, \ldots)/ \Q) \cong \prod_{p} \Z/2$$,

which can be deduced from the profinite limit

 * $$\cdots \to \operatorname{Gal}(\Q(\sqrt{2},\sqrt{3},\sqrt{5})/\Q) \to \operatorname{Gal}(\Q(\sqrt{2},\sqrt{3})/\Q) \to \operatorname{Gal}(\Q(\sqrt{2})/\Q)$$

and using the computation of the Galois groups.

Properties
The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory: the closed (with respect to the Krull topology) subgroups of the Galois group correspond to the intermediate fields of the field extension.

If $$E/F$$ is a Galois extension, then $$\operatorname{Gal}(E/F)$$ can be given a topology, called the Krull topology, that makes it into a profinite group.

title: Galois group