Hassane El

Fliouet
email: fliouet@yahoo.fr

ON THE PRESERVATION OF QUASI-MODULARITY

Keywords: 2010 Mathematics Subject Classication. Primary 12F15 Purely inseparable, Irrationality degree, Modular extension, w 0 -generated eld extensions, q-nite extension

published or not. The documents may come

Introduction

Let K/k be a purely inseparable eld extension of characteristic p ̸ = 0. Recall that K is modular over k if and only if for any (positive) integer n, K p n and k are linearly disjoint over K p n ∩ k. The notion of modular is to the purely inseparable theory what normal is to the separable theory. The preservation of modularity has been extensively studied, see for example ( [START_REF] Mordeson | Structure of arbitrary purely inseparable extension elds[END_REF], [START_REF] Rasala | Inseparable splitting theory[END_REF], [START_REF] Sweedler | Structure of inseparable extensions[END_REF], [START_REF] Waterhouse | The structure of inseparable eld extensions[END_REF]). In this vein, there exists smallest extensions denoted, respectively, by lm(K/k) and um(K/k) such that k -→ lm(K/k) -→ K -→ um(K/k) with K/lm(K/k) and um(K/k)/k are modular. Let us thus improve ( [START_REF] Fliouet | Generalization of quasi-modular extensions[END_REF], Theorem 3.3), we show that lm(K/k) ̸ = K when K/k is q-nite. More precisely, if K/k is of unbounded exponent, then K/lm(K/k) is also of unbounded exponent. However, if the irrationality degree of K/k is innite, it is highly probable that we lose this property by obtaining m = K. In the sequel, if [lm(K/k) : k] is nite, then K/k is called lq-modular (lower quasi-modular) extension and, if [um(K/k) : k] is nite, then K/k is called uq-modular (upper quasi-modular) extension (cf. [START_REF] Chellali | Extension presque modulaire[END_REF]). K/k is said to be q-modular (quasi-modular) if K/k is lower or upper quasimodular, it is the modularity up to nite extension. The main results about the q-modularity were proved by M. Chellali and the author in [START_REF] Chellali | Extension presque modulaire[END_REF] and [START_REF] Chellali | Théorème de la clôture lq-modulaire et applications[END_REF] under the assumption [k : k p ] < +∞. Namely, the q-modularity was characterized by means of invariants and its stability properties were investigated. Motivated by recent work published in [START_REF] Fliouet | Characterization of lq-modular extensions[END_REF] and [START_REF] Fliouet | Generalization of the lq-modular closure theorem and applications[END_REF] focusing on the generalization of some properties of lq-modularity, this paper continues the investigations begun in [START_REF] Chellali | Extension presque modulaire[END_REF] for an arbitrary eld k but with extra assumptions on K/k: the extension K/k needs to be q-nite, that is there must exist an integer M such that for every positive integer n the eld k p -n ∩ K is generated by at most M elements on k. We mainly study the stability of q-modularity with respect to inclusion and intersection, and consequently we are led to look closely at the properties of the minimal and maximal subextensions that respect this notion, as well as those of the q-modular closures. Similarly to the case of the purely inseparable extensions of nite degree [k : k p ], we show that the q-modularity is respected by any intersection covering k or K, i.e., for any family (k i ) of intermediate elds of q-nite extension K/k, we show that if each k i is q-modular (either lq-modular or uq-modular) over k, their intersection is q-modular over k and, if K is q-modular over each k i , it is q-modular over their intersection. As a result, there exists unique intermediate elds of K/k denoted, respectively, by lmq(K/k) and lumq(K/k) such that K/lmq(K/k) is lq-modular and K/lumq(K/k) is uq-modular. We emphasize in particular that lmq(K/k) is the relatively perfect closure of lm(K/k) and we give details about ulqm(K/K). However, the upper lq-(resp uq)-modular closures of q-nite extension K/k does not always exist. On the basis of certain considerations, we discuss in detail some conditions for the existence of these closures for a given q-nite extension. We also state that uq-modularity implies lq-modularity and that an important class of extensions are lq-modular.

Throughout this paper, unless otherwise stated, all considered elds are purely inseparable extensions of a common ground eld k. They are to be viewed as contained in a common algebraically closed eld Ω. For technical reasons, it will also be convenient to denote sometimes by [k, K] the set of intermediate elds of an extension K/k.

Preliminary

Recall that K is said to have an exponent (or, to be of bounded exponent) over k if there exists a positive integer e such that K p e ⊆ k, and the smallest integer that satises this relation will be called the exponent of K/k. Taking into account ( [START_REF] Mordeson | Structure of arbitrary purely inseparable extension elds[END_REF], Corollary 1.6), if K/k has an exponent, a subset B of K is an r-basis (relative p-basis) of K/k if and only if B is a minimal generating set of K/k. However, a minimal generating set may not exist in the general case (cf. [START_REF] Mordeson | Structure of arbitrary purely inseparable extension elds[END_REF], Lemma 1.16, Proposition 1.23). Let us consider a purely inseparable extension K/k of characteristic p > 0, clearly for any n ∈ N, k p -n ∩ K/k has an exponent, and in addition, the cardinality of any minimal generating set of k p -n ∩ K/k depends only on n.

Extending the minimum number of generator of K/k, due to M. F. Becker and S. Maclane in [START_REF] Becker | The minimum number of generators for inseparable algebraic extensions[END_REF], which was interesting/valid only in the case when K/k is nite, we have recently dened the irrationality degree of K/k as follows: di(K/k) = sup n∈N (|B n |) where |B n | is the cardinality of a minimal generating set B n of k p -n ∩ K over k (cf. [START_REF] Fliouet | Absolutely lq-nite extensions[END_REF], Denition 2.3), in which the sup is taken in the sense of ( [START_REF] Bourbaki | Eléments de Mathématique Théorie des ensembles[END_REF], III, p. 25, Proposition 2). If di(K/k) is nite, then K/k is called a q-nite extension (see [START_REF] Fliouet | Absolutely lq-nite extensions[END_REF], Denition 3.1), i.e., there must exist an integer M such that for every positive integer n the eld K ∩ k p -n is generated by at most M elements on k. It is clear that every nite purely inseparable eld extension is in particular q-nite. However the converse is true if and only if K/k has an exponent. We will often use the following theorem.

7). For any family

k ⊆ L ⊆ L ′ ⊆ K of purely inseparable extensions, we have di(L/L ′ ) ≤ di(K/k).
We will also need the notion of exponents. One of the fundamental theorems characterizing the exponents of a purely inseparable extension was published by Gunter Pickert [START_REF] Pickert | Inseparable Körperweiterungen[END_REF] in 1949, and can be stated as follows: If K/k is nite purely inseparable of multiplicity (irrationality degree) m, then there is an ordering of the generators, namely a 1 , . . . , a m , that the following conditions hold for i = 1, . . . , m:

(1) a i qi ∈ k(a 1 qi , . . . , a i-1 qi ), where q i = p ei and e i > 0.

(

) a i p -1 ̸ ∈ k(a 1 , . . . , a i-1 2
).

(3) e 1 ≥ e 2 ≥ . . . ≥ e m . Conversely, if K/k is generated by the m elements a 1 , . . . , a m satisfying the rst two conditions above, then the exponents e 1 . . . , e m are invariants of the extension. From now on, o i (K/k) denotes the i-th exponent of K/k (namely, o i (K/k) = e i if 1 ≤ i ≤ m, and by convention o i (K/k) = 0 for any integer i > m). For additional information on these invariants, we recommend referring to [START_REF] Chellali | Extensions purement inséparables d'exposant non borné[END_REF], [START_REF] Chellali | Sur les extensions purement inséparable[END_REF], [START_REF] Fliouet | Absolutely lq-nite extensions[END_REF], [START_REF] Rasala | Inseparable splitting theory[END_REF]. An r-basis (relative p-basis) {a 1 , a 2 , . . . , a m } of K/k is (preferably) called canonically ordered (Rasala uses in [START_REF] Rasala | Inseparable splitting theory[END_REF] the term normal generating sequence) if for j = 1, 2, . . . , m, o i (K/k) = o(a j /k(a 1 , . . . , a j-1 )) = o 1 (K/k(a 1 , . . . , a j-1 )), where o(x/k), for x ∈ K, is dened by o(x/k) = inf{j ∈ N| x p j ∈ k} (see for example [START_REF] Chellali | Sur les extensions purement inséparable[END_REF]). Note that any r-basis of K/k can canonically be ordered.

The relatively perfect closures of purely inseparable extensions also play an important role in this work. A eld k of characteristic p is said to be perfect if k p = k. In the same order of ideas, K/k is said to be relatively perfect if k(K p ) = K. We check immediately that:

If K/L and L/k are relatively perfect, then K/k is also perfect; If K/k is relatively perfect, then the same is true for L(K)/k(L); For any family (K i /k) i∈I of relatively perfect extensions,

∏ i K i /k is
(K/k) ⊆ K ∞ = ∩ k(K p i ). However, it is interesting to know when K ∞ /k is relatively perfect, or again rp(K/k) = K ∞ . First, if for some integer n, k(K p n )/k is relatively perfect, then for any integer s ≥ n, rp(K/k) = k(K p n ) = k(K p s ).
This condition holds in particular if K/k is q-nite or more generally if K/k(K p ) is nite. In the event that K/k is modular, Waterhause states in ( [START_REF] Waterhouse | The structure of inseparable eld extensions[END_REF], Proposition 4.2) that K ∞ is relatively perfect over k, and is the largest relatively perfect subextension of K.

Using relatively perfect closures, L.A. Kime in [START_REF] Kime | Purely inseparable modular extensions of unbounded exponent[END_REF] has studied the extension problem of Sweedler's theorem in the case of the innite exponent, and then gives a partial generalization of this theorem if K is of unbounded exponent over k. Here is one of those key results that we use in this paper frequently.

Theorem 2.2 ([13], Theorem 11). Any modular eld extension K over k, where for some nite

n, k(K p n )/k is relatively perfect, is isomorphic to ∩ i k(K p i ) ⊗ k M
where M is a modular subeld of k of nite exponent.

Stability of Lower quasi-modularity

Let P be the purely inseparable closure of k (inside an algebraic closure Ω of k). In [k : P ] we dene the relation ∼ as follows:

k 1 ∼ k 2 if and only if k 1 ⊆ k 2 and k 2 /k 1 is nite or k 2 ⊆ k 1 and k 1 /k 2 is nite.
We verify at once that ∼ is reexive, symmetric, however ∼ is generally nontransitive. Moreover, for any q-nite extension K 1 /k, the application of lower modularity:

lm : [k : K 1 ] -→ [k : K 1 ] L -→ lm(K 1 /L),
is compatible with the relation ∼. More specically, we have:

Proposition 1. Let k 1 ⊆ k 2 ⊆ K 1 be q-nite extensions. If k 1 ∼ k 2 , then lm(K 1 /k 1 ) ∼ lm(K 1 /k 2 ). Proof. It is enough to note that lm(K 1 /k 1 ) ⊆ lm(K 1 /k 2 ), and if moreover o 1 (k 2 /k 1 ) = e 1 , then k 2 ⊆ (lm(K 1 /k 1 )) p -e 1 ∩ K 1 with K 1 /(lm(K 1 /k 1 )) p -e 1 ∩ K 1 is modular, (cf. [5], Proposition 6.4). Let so lm(K 1 /k 2 ) ⊆ (lm(K 1 /k 1 )) p -e 1 ∩ K 1 with (lm(K 1 /k 1 )) p -e 1 ∩ K 1 /lm(K 1 /k 1 ) is nite, since K 1 /k is q-nite.
As a consequence, the lq-modularity is stable up to a nite extension of the choice of the ground eld, as it is stated more generally by the following result. Proposition 2. Let K/k be a q-nite extension. We have the following properties:

(1

) If k ′ ∼ k and k ′ ⊂ K, K/k is lq-modular if and only if the same is true for K/k ′ . (2) If K ∼ K ′ and k ⊂ K ′ , K/k is lq-modular if and only if K ′ /k is also lq-modular. (3) If k ′ ∼ k and K ∼ K ′ with k ′ ⊂ K ′ , then K/k is lq-modular if and only if the same is true for K ′ /k ′ .
Proof. For a proof cf. [START_REF] Fliouet | Characterization of lq-modular extensions[END_REF], Proposition 4.8

As a consequence, the result below makes it possible to reduce the study of lq-modularity to the case of relatively perfect extensions.

Corollary 3.1. Let K/k be a q-nite extension. Then the following assertions hold true:

(i) K/k is lq-modular if and only if the same is true for rp(K/k)/k. (ii) For every F ∈ [k : K], K/F is lq-modular if and only if the same is true for rp(K/k)/rp(F/k) and de K/rp(F/k). Proof. Just notice that K ∼ rp(K/k) and F ∼ rp(F/k).
Let K/k be a purely inseparable extension and k 0 the maximal perfect subeld of k. It is clear that K/k 0 is modular (and in particular lq-modular). So we can hope that K/k is lq-modular! In other words, any purely inseparable extension would be lq-modular, however K/k 0 is transcendent. Here is a non-obvious example of a purely inseparable extension that is not lq-modular.

First, we set notations which will be used in the sequel of the paper. For a subset C of K, we let C p -∞ denote the set {a p -i | a ∈ C, i = 1, 2 . . .}, and let a p -∞ denote {a} p -∞ .

Example 1. We return to ([9], example 4.5). Let (X, Z 1 , Z 2 ) be independent indeterminates over

k 0 and k = k 0 (X, Z 1 , Z 2 ). For all n ∈ N * , put K n = k(X p -2n , θ n ), with θ 1 = Z p -1 1 X p -2 + Z p -1 2

, and for every integer

n ≥ 2, θ n = Z p -1 1 X p -2n + (θ n-1 ) p -1 = Z p -1 1 X p -2n + Z p -2 1 X p -2n+1 + . . . + Z p -n 1 X p -n-1 + Z p -n 2 . Clearly θ p n+1 = Z 1 X p -2n-1 + θ n , so K n ⊂ K n+1 , and hence K = ∪ n K n is a commutative eld. Under the conditions above, k(X p -∞ )/k is the unique minimal intermediate eld of K/k such that K/k(X p -∞ ) is modular (lm(K/k) = k(X p -∞ )).
Namely the proof of this theorem uses the following result which is an immediate consequence of the modularity criterion. This result is basic and will be used repeatedly, usually without explicit quotation. Lemma 3.2 ([3], Lemma 3.7). Let K/k be a modular extension of characteristic p ̸ = 0 and

((a, b), (e 1 , e 2 )) element of k 2 × K 2 such that e 2 p j = ae 1 p j + b (j being a positive integer). If e 1 p j ̸ ∈ k, then a p -j and b p -j belong to K.
Proof. Notice that e 1 p j ̸ ∈ k is identically equivalent to (1, e 1 p j ) is linearly independent over k, and in particular, (1, e 1 p j ) is linearly independent over k ∩ K p j . Let's extend this system into a linear basis B of K p j over k ∩ K p j . As K p j and k are k ∩ K p j -linearly disjoint (K/k is modular), B is also a linear basis of k(K p j ) over k. Since e 2 p j = ae 1 p j + b with a and b belong to k and e 2 p j is written uniquely as a sum of elements of B, then by identication we will have a

∈ k ∩ K p j and b ∈ k ∩ K p j . It means that a p -j ∈ K and b p -j ∈ K.
The lq-modularity is preserved not only under nite extension but even under any extension of the ground eld. Proposition 3. Let K/k be a q-nite extension. For every L ∈ [k : K], if K/k is lq-modular, then the same is true for K/L.

Proof. From Proposition 2, it is sucient to present the proof when K/k and L/k are relatively perfect. Since

k(L p n ) ⊂ k(L ∩ K p n ) ⊂ L, therefore L = k(L ∩ K p n ), and consequently k(K p n ) = K and L are linearly disjoint over k(L ∩ K p n ).
We thus nd ourselves under the conditions of ( [START_REF] Mordeson | Structure of arbitrary purely inseparable extension elds[END_REF], Lemma 1.60), which gives us K/L is modular if K/k is modular. Otherwise there exists an intermediate eld k 1 of nite degree over k such that K/k 1 is modular, so the same is true for

K/k 1 (L), since k 1 (L)/k 1 is relatively perfect. As k 1 (L)/L is nite, then K/L is lq-modular. Theorem 3.3. Let K/k be a q-nite extension and (k i ) i∈I a family of intermediate elds of K/k. If K is lq-modular over each k i , it is lq-modular over their intersection.
Proof. Thanks to ([12], Proposition 2.5), it suces to prove this result for I = {1, 2}. Firstly, for simplicity denote lm(K/k j ) by m j for j = 1, 2. Taking into account the lq-modularity, there exists a positive integer e such that m j ⊆ k j p -e ∩K for j = 1, 2; and hence

m 1 ∩m 2 ⊆ k 1 p -e ∩k 2 p -e ∩K = (k 1 ∩ k 2 ) p -e ∩ K. It follows that m 1 ∩ m 2 /k is nite (namely K/k is q-nite). On the other hand, by virtue of ([19], Proposition 1.2), K/m 1 ∩ m 2 is modular, so K/k 1 ∩ k 2 is lq-modular.
As a result, we have: Corollary 3.4. For every q-nite extension K/k, there exists an unique minimal intermediate eld m of K/k such that K/m is lq-modular.

Proof. Immediate.

Henceforth, we denote by lqm(K/k) the minimal intermediate eld of K/k over which K is lq-modular. Clearly lm(K/k) ⊆ lqm(K/k), but we can now even better situate lqm(K/k).

Proposition 4. Under the notations above we have we have lqm(K/k) = rp(lm(K/k)/k). In particular, lqm(K/k)/k is relatively perfect.

Proof. Let m 0 = lqm(K/k), m 1 = lm(K/m 0 ) and m 2 = lm(K/k). It is clear that m 0 ⊆ m 2 ⊆ m 1 and m 1 /m 0 is nite, since K/m 0 is lq-modular. Whence rp(m 0 /k) = rp(m 2 /k). But, according to Corollary 3.1, K/rp(m 0 /k) is also lq-modular, so m 0 = rp(m 0 /k) = rp(m 2 /k).
This leads to: Corollary 3.5. Let K and K ′ be two intermediate elds of q-nite extension K/k. We have the following assertions:

(i) If K ∼ K ′ , then lqm(K/k) = lqm(K ′ /k). In particular, lqm(K/k) = lqm(rp(K/k)/k). (ii) For every L ∈ [k : K], L(lqm(K/k)) = lqm(K/L).
Proof. Taking into account Proposition 2, it suces to prove assertion

(ii). Let m 1 = lqm(K/k) and m 2 = lqm(K/L). Clearly m 1 ⊆ m 2 , therefore m 1 (L) ⊆ m 2 . But, according to Proposition 3, K/m 1 (L) is lq-modular, so m 2 ⊆ m 1 (L). Whence m 1 (L) = m 2 .
The preservation of lq-modularity by intersection seems much less obvious. First, we will need, as in the nite case [k : k p ] < ∞ (cf. [START_REF] Chellali | Extensions i-Modulaires[END_REF], [START_REF] Chellali | Extension presque modulaire[END_REF]), the following results.

Let K 1 and K 2 be two intermediate elds between k and K that are k-linearly disjoint. The following result is a well known consequence of the transitivity of linear disjointness. Proposition 5. For every

L 1 ∈ [k, K 1 ] and L 2 ∈ [k, K 2 ], L 2 (K 1 ) and L 1 (K 1 ) are k(L 1 , L 2 )- linearly disjoint. In particular, L 2 (K 1 ) ∩ L 1 (K 2 ) = k(L 1 , L 2 ). Corollary 3.6. For any subset G of K 2 such that K 1 (K 2 ) = K 1 (G), we have K 2 = k(G). In addition, if G is a linear basis of K 1 (K 2 ) over K 1 , then G is also linear basis of K 2 over k.
Consider now two intermediate elds L and M of nite degree over k of a purely inseparable extension K/k. Let B = (α 1 , α 2 , . . . , α n ) be a canonically ordered r-basis of L/k. Let e i be the i-th exponent of L/k (e i = o i (L/k)) and e the exponent of M/k (e = o 1 (M/k)). We assume that there exists an integer s ∈ [1, n] such that e s-1 -e s > e. According to ([6], Proposition 9), there are unique constants a ξ ∈ k(L p es +1 ) such that

α p es s = ∑ ξ∈Λ a ξ (α 1 , α 2 , . . . , α s-1 ) ξp es , where Λ = {(i 1 , i 2 , . . . , i s-1 )| 0 ≤ i j ≤ p -1}.
Suppose further that K p es and k(M ∩ K p es ) are M ∩ K p es -linearly disjoint. It turns out that this condition holds as soon as K/M is modular.

Lemma 3.7. Under the assumptions above, we have:

∀ξ ∈ {(i 1 , i 2 , . . . , i s-1 )| 0 ≤ i j ≤ p -1} , a p -es ξ ∈ k p -e 1 +1 ∩ K.
Proof. First, we show that ((α 1 , α 2 , . . . , α s-1 ) ξp es ) ξ∈Λ is a linear basis of k(M ∩ K p es )(α 1 p es , . . . ,

α s-1 p es ) over k(M ∩ K p es )(α 1 p es +1 , . . . , α s-1 p es +1 ).
For that, it suces to show that

di(k(M ∩ K p es )(α 1 p es , . . . , α s-1 p es )/k(M ∩ K p es )) = s -1.
If there exists i ∈ {1, . . . , s-1} such that α i p es belongs to k(M ∩K p es )(α

It follows from ([6]

, Lemma 1) that o s-1 (L/k) = o s-1 (k(α 1 , . . . , α s-1 )/k) = e s-1 ≤ e s +e, or again e s-1 -e s ≤ e, a contradiction. Moreover, we have K p es and k(M ∩K p es ) are M ∩K p es -linearly disjoint, so in particular M ∩K p es (L p es ) and k(M ∩K p es ) are M ∩K p es -linearly disjoint. As k(L p es ) = k(α 1 p es , . . . , α s-1 p es ), therefore especially k(M ∩ K p es )(L p es ) = k(M ∩ K p es )(α 1 p es , . . . , α s-1 p es ). According to the previous Corollary M ∩ K p es (L p es ) = M ∩ K p es (α 1 p es , . . . , α s-1 p es ), and consequently α s p es ∈ M ∩ K p es (α 1 p es , . . . , α s-1 p es ). On the other hand, by transitivity of linear disjointness, M ∩ K p es (L p es ) and k(M ∩ K p es )(L p es +1 ) are M ∩ K p es (L p es+1 )-linearly disjoint. Since

B 1 = ((α 1 , α 2 , . . . , α s-1 ) ξp es ) ξ∈Λ is a linear basis of k(M ∩ K p es )(L p es ) over k(M ∩ K p es )(L p es+1 ), then B 1 is also linear basis of M ∩ K p es (L p es ) over M ∩ K p es (L p es +1 ). But α p es s = ∑ ξ a ξ (α 1 , α 2 , . . . , α s-1 ) ξp es with a ξ ∈ k(L p es+1 ) ⊆ k(M ∩K p es )(L p es +1 ); by identication, for any ξ ∈ Λ, a ξ ∈ M ∩K p es (L p es +1 ) ⊆ K p es . It follows that a ξ p -es ∈ K. We also have a ξ ∈ k(L p es +1 ) ⊆ k p -e 1 +es+1 (namely, o 1 (k(L p es +1 )/k) = e 1 -e s -1). Whence a ξ ∈ k p -e 1 +1
, and so a ξ ∈ k p -e 1 +1 ∩ K. Theorem 3.8. Let (K i /k) i∈I be a family of lq-modular intermediate elds of a q-nite extension K/k. Then

∩ i∈I K i /k is lq-modular.
Proof. Thanks to ([12], Proposition 2.5), we reduce to I = {1, 2}. We have K 1 /k and K 2 /k are lqmodular, so according to ([9], Theorem 4.6) and ([5], Proposition 6.4), there exists a positive integer e such that K 1 /k p -e ∩ K 1 and K 2 /k p -e ∩ K 2 are modular. The q-simple case (i.e. di(K j /k) = 1, for j = 1, 2) and the nite case are trivially obvious, so it is sucient to establish the result when

K 1 ∩ K 2 /k is of unbounded exponent and of irrationality degree 1 < di(K 1 ∩ K 2 /k). It means that rp(K 1 ∩ K 2 /k) is not trivial (rp(K 1 ∩ K 2 /k) ̸ = k, or again 1 ≤ di(rp(K 1 ∩ K 2 /k)/k))
and, for every positive integer j, k p -j ∩ (K 1 ∩ K 2 ) has an exponent j over k. In the sequel, we use the following notations:

t = di(rp(K 1 ∩ K 2 )/k), U j s (K 1 ∩ K 2 /k) = j -o s (k p -j ∩ K 1 ∩ K 2 /k) for any integer s ∈ [1, t], e j s = o s (k p -j ∩ K 1 ∩ K 2 /k), and ε j s = e j+1
s -e j s (ε j s ∈ {0, 1}). Let i 0 denote the smallest positive integer for which the sequence

(U j i0 (K 1 ∩ K 2 /k)) j∈N is unbounded. If i 0 = t + 1, by virtue of ([9], Theorem 4.6), K 1 ∩ K 2 /k is lq-modular. Suppose then that 1 < i 0 ≤ t.
From the minimality property of i 0 , we conclude that lim

j-→+∞ (U j i0 (K 1 ∩ K 2 /k)) = +∞, and for each integer s ∈ [1, i 0 -1], the increasing sequence of integers (U j s (K 1 ∩ K 2 /k))
j∈N is bounded, therefore stationary. Whence there exists a positive integer e 1 , for every integer j ≥ e 1 , for every

integer s ∈ [1, i 0 -1], U j s (K 1 ∩ K 2 /k) = U j+1 s (K 1 ∩ K 2 /k). Furthermore, lim j-→+∞ (U j i0 (K 1 ∩ K 2 /k) - U j i0-1 (K 1 ∩ K 2 /k)) = lim j-→+∞ (e j i0-1 -e j i0 ) = +∞.
As a result, there exists n 0 > e 1 , for every

n ≥ n 0 , e n i0-1 -e n i0 >>> e. If ε n i0 = 1 for each n ≥ n 0 , then the sequence (U n i0 (K 1 ∩ K 2 /k)) n∈N is bounded, a contradiction. Whence there exists n 1 ≥ n 0 such that ε n1 i0 = 0, or again e n1 i0 = e n1+1 i0
. Let n 2 be the largest integer such that e n2 i0 = e n1 i0 (n 2 exists, because 1 < i 0 ≤ t, and so by ( [START_REF] Fliouet | Absolutely lq-nite extensions[END_REF], Theorem 3.9), lim n-→+∞ (e n i0 ) = +∞). According to ( [START_REF] Chellali | Extensions purement inséparables d'exposant non borné[END_REF], Proposition 8.4), there exists a canonically ordered r-basis {α ) and Λ = {(i 1 , i 2 , . . . , i i0-1 )| 0 ≤ i j ≤ p -1}. As

1 , . . . , α m } of k p -n 2 -1 ∩ K 1 ∩ K 2 /k, there exists ε i0+1 ∈ {1, p}, . . . , ε m ′ ∈ {1, p}, (m ′ = di(k p -n 2 ∩K 1 ∩K 2 /k)), such that {α 1 p , . . . , α i0 p , α i0+1 εi 0 +1 , . . . , α m ′ ε m ′ } is also a canonically ordered r-basis of k p -n 2 ∩ K 1 ∩ K 2 /k.
K 1 /k p -e ∩ K 1 and K 2 /k p -e ∩ K 2 are modular, by virtue of Lemma 3.7, a ξ p -e n 2 +1 i 0 ∈ k p -n 2 ∩ K 1 and a ξ p -e n 2 +1 i 0 ∈ k p -n 2 ∩ K 2 .
According to the denition equation α i0 , we also deduce that

α i0 belongs to k(α 1 , . . . , α i0-1 , ((a ξ ) p -e n 2 +1
i 0

) ξ ), and hence

e n2 i0 = e n2+1 i0 -1 = o(α i0 p , k(α 1 p , . . . , α i0-1 p )) ≥ o 1 (k(α 1 p , . . . , α i0-1 p , ((a ξ ) p -e n 2 +1 i 0 ) ξ )/k(α 1 p , . . . , α i0-1 p )) ≥ o 1 (k(α 1 , . . . , α i0-1 , ((a ξ ) p -e n 2 +1 i 0 ) ξ )/k(α 1 , . . . , α i0-1 )) ≥ o(α i0 , k(α 1 , . . . , α i0-1 )) = e n2+1 i0 (cf. [6], Proposition 6). As a result, e n2+1 i0 -1 ≥ e n2+1 i0
, a contradiction.

In ( [START_REF] Sweedler | Structure of inseparable extensions[END_REF], p. 408), Sweedler has shown that any purely inseparable extension K over k of nite exponent is contained in a unique minimal eld extension which we denoted earlier by um(K/k), where um(K/k) is modular over k; um(K/k) is called the modular closure of K over k. L.A. Kime has extended um(K/k) to the innite exponent case (cf. [START_REF] Kime | Purely inseparable modular extensions of unbounded exponent[END_REF], Theorem 4). It is not true, however, that a q-nite extension K/k admit minimal lq-modular extensions over k, as we will in the following counterexample. If we restrict ourselves to the case where [k : k p ] is nite (or also k p -∞ /k is q-nite), then any purely inseparable extension of k is q-nite. In particular, the modular closure um(K/k) of K/k is q-nite, so the set of purely inseparable extensions of K that are lq-modular over k is not empty, and by Theorem 3.8, there exists an unique minimal extension

M of K such that M/k is lq-modular. Example 2 (Counterexample). Given a perfect eld k 0 of characteristic p > 0 and {X, Y 1 , Z i : i ∈ N * } a family of independent indeterminates over k 0 . Let k = k 0 (X, Y 1 , (Z i ) i∈N * ) and K = k(X p -∞ , θ 1 , θ 2 , . . .), where θ 1 = Z 1 p -1 X p -2 + Y 1 p -1 and, for each i ≥ 2, θ i = Z i p -1 X p -2i + (θ i-1 ) p -1 .
We immediately verify that, for any integer i ≥ 2,

θ i = Z i p -1 X p -2i + (θ i-1 ) p -1 , = Z i p -1 X p -2i + • • • + Z 1 p -i X p -i-1 + Y 1 p -i .
In particular, for each positive integer

i, k(X p -∞ )(θ i ) = k(X p -∞ )(θ i+1 p ), and so k(X p -∞ , θ i ) has an exponent i over k(X p -∞ ). Proposition 6. The family (Z i ) i∈N * is p-independent in K.
Proof. Suppose there exists a positive integer i > 1 such that

Z i p -1 ∈ K(Z 1 p -1 , . . . , Z i-1 p -1 ) if i ̸ = 1 and, where appropriate, Z 1 p -1 ∈ K. As θ i = Z i p -1 X p -2i + • • • + Z 1 p -i X p -i-1 + Y 1 p -i , or again Y 1 p -i = θ i -Z i p -1 X p -2i + • • • + Z 1 p -i X p -i-1 , we conclude that Y 1 p -1 belongs to K(Z 1 p -1 , . . . , Z i-1 p -1
), and hence k(

X p -1 , Z 1 p -1 , . . . , Z i p -1 , Y 1 p -1 ) ⊆ K(Z 1 p -1 , . . . , Z i-1 p -1
).

By virtue of Theorem 2.1 and (

[8], Corollary 2.5), i + 2 = di(k(X p -1 , Z 1 p -1 , . . . , Z i p -1 , Y 1 p -1 )/k), ≤ di(K(Z 1 p -1 , . . . , Z i-1 p -1 )/k), ≤ di(K/k) + di(k(Z 1 p -1 , . . . , Z i-1 p -1 )/k) = 2 + i -1 = i + 1, a contradiction. It follows that Z 1 p -1 ̸ ∈ K and, for every i > 1, Z i p -1 ̸ ∈ K(Z 1 p -1 , . . . , Z i-1 p -1
),

which is equivalent to Z 1 ̸ ∈ K p and, for every i > 1, Z i ̸ ∈ K p (Z 1 , . . . , Z i-1 ). According to the exchange property, (Z i ) i∈N * is p-independent in K.

Proposition 7. Under the conditions above, for any q-nite extension F/K, we have k(X p -∞ ) ⊆ lm(F/k). In addition, k(X p -∞ ) is the unique minimal intermediate eld of K/k over which K is modular (lm(K/k) = k(X p -∞ )). In particular, K/k is not lq-modular.

Proof. It is immediately that K/k(X p -∞ ) is modular and, as the q-nitude is transitive, then F/k is q-nite. Let m = lm(F/k). Suppose there exists a positive integer n such that X p -n+1 ∈ m, but X p -n ̸ ∈ m. Now, let j be a nonzero natural number. Since θ n+j is dened as follows:

θ n+j = Z n+j p -1 X p -2(n+j) + Z n+j-1 p -2 X p -2(n+j)+1 + • • • + Z 1 p -n-j X p -n-j-1 + Y 1 p -n-j , so θ n+j p n+2j = Z n+j p n+2j-1 X p -n + Z n+j-1 p n+2j-2 X p -n+1 + • • • + Z 1 p j X p j-1 + Y 1 p j . Notice that Z n+j p n+2j-1 and Z n+j-1 p n+2j-2 X p -n+1 + • • • + Z 1 p j X p j-1 +Y 1 p j belonging to k ⊆ m, so by Lemma 3.2, for every j ∈ N * , Z n+j p -1 ∈ m p -n-2j ∩ F ⊆ F . It follows that, k(Z n+1 p -1 , Z n+2 p -1 , . . .) ⊆ F . Since (Z i ) i∈N * is p-independent in k, then by Theorem 2.1, |{Z n+1 p -1 , Z n+2 p -1 , . . .}| = di(k(Z n+1 p -1 , Z n+2 p -1 , . . .)/k) ≤ di(F/k) < ∞,
a contradiction. Accordingly, lm(K/k) = k(X p -∞ ), and so K/k is not lq-modular.

Let N/k be the union of chain (totally ordered family under inclusion) of purely inseparable extensions (N j ) j∈I . Taking into account the minimality of modular closures, for every j ∈ I, um(N j /k) ⊆ um(N/k). But according to ([19], Proposition 1.2), ∪ j∈I um(N j /k) is modular over k, so N ⊆ ∪ j∈I um(N j /k), and consequently um(N/k) = ∪ j∈I um(N j /k). This makes it possible to reduce the study of modular closures to the case of nite extensions, in which case the Rasalat's process can be used eectively to determine um(N/k) (cf. [START_REF] Rasala | Inseparable splitting theory[END_REF]). In general, the modular closure of a purely inseparable extension can't be precisely located. However, it is known that if N/k has an exponent e, then um(N/k)/k has also the same exponent e. If moreover N/k is nite, um(N/k) is also nite over k. So, we can hope that the q-nitude will be respected by passing to the modular closures, i.e., if N/k is q-nite, the same is true for um(N/k)/k. However, [k : k p ] is not nite or k p -∞ /k is not always q-nite!!! Proposition 8. Under the notations described above,

k p -∞ = k(X p -∞ , Y 1 p -∞ , Z 1 p -∞ , Z 2 p -∞ , . . .) is the modular closure of K/k.
Proof. We proceed by induction using the same techniques that were used in the proof of the previous Proposition. First, it is obvious that k p -∞ /k is a modular extension that contains K. For simplicity denote um(K/k) by M . Recall that

θ 1 = Z 1 p -1 X p -2 + Y 1 p -1 and, for all integer j ≥ 2, θ j = Z j p -1 X p -2j + Z j-1 p -2 X p -2j+1 + . . . + Z 1 p -j X p -j-1 + Y 1 p -j , so θ 1 p = Z 1 X p -1 + Y 1 and, for j ≥ 2, θ j p 2j-1 = Z j p 2j-2 X p -1 + Z j-1 p 2j-3 X + . . . + Z 1 p j-1 X p j-2 + Y 1 p j-1 . Since Z 1 , Y 1 , Z j p 2j-2 , and Z j-1 p 2j-3 X + • • • + Z 1 p j-1 X p j-2 + Y 1 p j-1 belongs to k, then according to Lemma 3.2, we get Y 1 p -1 and Z j p -1
∈ M for each positive integer j. Let i be an integer ≥ 2, assume that Y 1 p -i and Z j p -i belongs to M for each positive integer j. We now denote Z j p

-i-1 X p -2(j+i)+i + . . . + Z 1 p -j-i X p -j-i-1 + Y 1 p -j-i by β j . As θ j+i = Z j+i p -1 X p -2(j+i) + Z j+i-1 p -2 X p -2(j+i)+1 + . . . + Z j p -i-1 X p -2(j+i)+i + . . . + Z 1 p -j-i X p -j-i-1 + Y 1 p -j-i , then θ j+i can be written as θ j+i = Z j+i p -1 X p -2(j+i) + Z j+i-1 p -2 X p -2(j+i)+1 + . . . + Z j+1 p -i X p -2(j+i)+i-1 + β j with Z j+i p -1 X p -2(j+i) + Z j+i-1 p -2 X p -2(j+i)+1 + . . . + Z j+1 p -i
X p -2(j+i)+i-1 belongs to M . In particular, β j ∈ M for every positive integer j. We also have β 1

p i+1 = Z 1 X p -1 + Y 1 and, for j ≥ 2, β j p 2j+i-1 = Z j p 2j-2 X p -1 + . . . + Z 1 p j-1 X p j-2 + Y 1 p j-1 with Z j p 2j-2 and Z j-1 p 2j-3 X + . . . + Z 1 p j-1 X p j-2 + Y 1 p j-1 belongs to k.
Similarly to the previous case, we will then have

Y 1 ∈ k ∩ M p i+1 , Z 1 ∈ k ∩ M p i+1 , and Z j p 2j-2
∈ k ∩ M p 2j+i-1 for every j ≥ 2, therefore Y 1 p -i-1 and Z j p -i-1 belongs to M for all positive integer j. Hence the result is obtained by induction.

In the sequel, given a positive integer n, for every j ∈ N * , let

α j n = θ n p -j , = Z n p -1-j X p -2n-j + • • • + Z 1 p -n-j X p -n-j-1 + Y 1 p -n-j , and L j n = k(X p -∞ , ((Z i p -∞ ) i>n , θ n+j ). Since θ n+j = Z n+j p -1 X p -2(i+j) + θ n+j-1 p -1 , = Z n+j p -1 X p -2(i+j) + • • • + Z n+1 p -j X p -2n-j-1 + θ n p -j , = Z n+j p -1 X p -2(i+j) + • • • + Z n+1 p -j X p -2n-j-1 + α j n ; so L j n = k(X p -∞ , (Z i p -∞ ) i>n , θ n+j ) = k(X p -∞ , (Z i p -∞ ) i>n , α j n ) for all j ∈ N * . Lemma 4.2. Let K/k be a q-nite extension. Every minimal intermediate eld m 1 of K/k over which K is uq-modular is contained in rp(K/k). Proof. Let m 2 = lm(m 1 (rp(K/k))/rp(K/k)). Suppose that m 1 (rp(K/k)) ̸ = rp(K/k), by ([10], Theorem 3.3), m 2 ̸ = m 1 (rp(K/k)).
Since K/m 1 is uq-modular, then according to Proposition 10, m 1 (rp(K/k))/m 1 is modular, and consequently m 1 (rp

(K/k))/m 1 ∩ m 2 is modular. It follows that K/m 1 ∩ m 2 is uq-modular. But m 1 /k is minimal, so m 1 ∩ m 2 = m 1 ⊆ m 2 . Whence m 1 (rp(K/k)) ⊆ m 2 (rp(K/k)) = m 2 ⊆ m 1 (rp(K/k)), or again m 2 = m 1 (rp(K/k)), a contradiction. We then deduce that m 1 (rp(K/k)) = rp(K/k), that is, m 1 ⊆ rp(K/k).
Proof Which implies that m ⊆ m 1 ⊆ L. But K/m is also uq-modular, we deduce that m 1 = m, and therefore m is the only minimal intermediate eld of K/k such that K/m is uq-modular.

Corollary 4.3. Let K/k be a q-nite extension, {k i } is a family of intermediate elds. If K is uq-modular over each k i , it is uq-modular over their intersection.

Unlike the case when [k : k p ] is nite, in which any extension of k is q-nite, a q-nite extension K/k may not have minimal uq-modular extensions over k, as Example 2 (counterexample) demonstrate. In the sequel, the smallest extension M/K such that M/k is uq-modular when it exists will be called the uq-modular closure of K/k and will be denoted by ulqm(K/k). However, the result which follows gives a necessary and sucient condition for the existence of the uq-modular closure. More specically, we have: Proposition 13. Given a q-nite extension K/k and M = um(K/k). Then K/k admits an uqmodular closure if and only if [M : rp(M/K)] is nite. Moreover, when ulqm(K/k) exists, then ulqm(K/k) = rp(M/K).

Proof. It is immediate that if M/rp(M/K) is nite, then rp(M/K)/k is uq-modular. Let K 1 /K be a purely inseparable extension such that K 1 /k is uq-modular, so K 2 = um(K 1 /k) has an exponent e over K 1 . As a result, M ⊆ K 2 , and in particular, K(M p e ) ⊆ K(K 2 p e ) ⊆ K 1 . Consequently, rp(M/K) ⊆ K 1 , and therefore ulqm(K/k) = rp(M/K).

Conversely, we denote uqlm(K/k) by N and um(N/k) by N 1 . It is clear that N ⊆ M ⊆ N 1 and M/N is nite. On the other hand, N/K is relatively perfect. Otherwise, let G be an r-basis of N/K and choose an element a from G. As [N 1 :

K(N p )(G \ {a})] = [N 1 : N ][N : K(N p )(G \ {a})] = p[N 1 : N ],
we deduce that K(N p )(G \ {a}) is uq-modular and is better than N , a contradiction. Since We can weaken condition [M : rp(M/K)] is nite by [M : k(M p )] is nite, or again M/rp(M/k) is nite. In fact, as K/k is q-nite, then

Theorem 2 . 1 (

 21 [START_REF] Fliouet | Absolutely lq-nite extensions[END_REF], Theorem 2.

  [M : K(M p ] ≤ [M : K(N p )] = [M : N ] < ∞, therefore rp(M/K) = rp(N/K) = N and M/rp(M/K) is nite, this means that rp(M/K) = uqlm(K/k).

  [N : k(N p )] = [K(N p ) : k(N p )] ≤ [K : k(K p )] < ∞, and therefore [M : k(M p )] ≤ [M : N ][N : k(N p )] < ∞.

  also relatively perfect.

	Therefore, there exists an unique maximal intermediate eld M of K/k such that M/k is relatively
	perfect (for more details see [16], Proposition 6). M is called the relatively perfect closure of K/k
	and is denoted by rp(K/k). Moreover, the relatively perfect veries the associativity-transitivity
	property, i.e., for every L ∈ [k : K], we have rp(rp(K/L)/k) = rp(K/k) and rp(K/rp(L /k)) =
	rp(K/k). In particular, the relatively perfect closure remains invariant up to extension of nite
	exponent, i.e., if K/L is of nite exponent, then rp(K/k) = rp(L/k), and if, in addition, K/k is
	relatively perfect, then K/L is of nite exponent implies that L = K. Generally, we can not exactly
	locate rp(K/k), all we can say is that rp

  The denition equation of α i0 allows to write

			α i0	p	e n 2 +1 i 0	=	∑	a ξ (α 1 , . . . , α i0-1 ) ξp	e n 2 +1 i 0	,
									ξ
	where a ξ ∈ k(α 1 p e n 2 +1 i 0	+1	, . . . , α i0-1	p	e n 2 +1 i 0	+1

  . (Proof of Theorem 4.1) Let L be an intermediate eld of K/k such that K/L is uq-modular and m = lm(rp(K/k)/k). We denote by H the set of intermediate elds of L/k over which K is uq-modular. The set H ordered by inverse inclusion is inductive (Hint:K 1 ≤ K 2 if and only if K 2 ⊆ K 1 ). Indeed, • H ̸ = ∅, since m ∈ H. • Let (K i ) i∈I bea totally ordered family in H. According to ([12], Proposition 2.5), there exists i 0 ∈ I such that K i0 = ∩ i∈I (K i ) and K i0 /k is maximal. According to Zorn's Lemma, H has a maximal element (for inverse inclusion) that we denote by m 1 . In other words, m 1 is a minimal intermediate eld of K/k over which K is uq-modular. Whence according to Lemma 4.2, m 1 ⊆ rp(K/k) and, by Proposition 10, rp(K/k)/m 1 is modular.

Proposition 9. For every positive integer j, we have

In particular, L j n /k(X p -n ) is modular.

Proof. It is sucient to show that for every positive integer j,

) i>n )).

We immediately verify that (α j n )

), and therefore Y 1

and so

It follows that

) i>n ) for every n ≥ 1. We immediately verify that

.).

Theorem 3.9. For every positive integer n, K n /k is an lq-modular extension satisfying the following intersection property

Moreover, K/k does not admit minimal lq-modular extensions over k.

Proof. We return to the notation

n is also modular over k(X p -n ), and a fortiori K n /k is lq-modular. Choose an element λ of ∩ K n , therefore for every positive integer n, λ belongs to K n . In particular, λ belongs to K 1 , and consequently there exists a positive integer

By virtue of transitivity of linear disjointness,

)∩K i = K, and consequently

) and K i are K-linearly disjoint. This result corrects Corollary 3.14.1 in [START_REF] Fliouet | Extension de la i-modularié[END_REF] which claim that, any q-nite extension K over k is contained in a unique minimal eld extension L, where L is lq-modular over k. A rst cause of this error comes from the fact that the set of lq-modular extensions over k containing K can be empty.

Stability of upper quasi-modularity

Recall that K/k is uq-modular (upper quasi-modular) if the modular closure um(K/k) of K is a nite extension of K. The following result allows us to reduce the study of uq-modularity to relatively perfect extensions. Proposition 10. A q-nite extension K/k is uq-modular if and only if rp(K/k)/k is modular.

The converse part of the proof uses results due to Theorem 2.2. Indeed, if e denotes the exponent of K/rp(K/k), as an immediate consequence of modularity, S = (rp(K/k)) p -e is modular over k. By virtue of Theorem 2.2, there exists an r-basis of S/k such that

In the class of q-nite extensions, the property of uq-modularity is ner than that of lqmodularity. More precisely, we have: Proposition 11. Let K/k be a q-nite extension. If K/k is uq-modular, then K/k is lq-modular.

Proof. Follows from Corollary 3.1, since K ∼ rp(K/k) for any q-nite extension K/k.

But the converse is usually false.

It's clear that K/k(X p -1 ) is modular, therefore lq-modular, but K/k is not uq-modular. Indeed, if K/k is uq-modular, by Proposition 10,

As in the case of lq-modularity, the uq-modularity is compatible with the relation ∼. In addition, the uq-modularity is stable up to nite extension, as shown by the following result. Proposition 12. Let K and K ′ be two intermediate elds of a q-nite extension H/k. If K ∼ K ′ , then K/k is uq-modular if and only if the same is true for K ′ /k.

Proof. Immediately follows from Proposition 10 and the fact that rp(K/k) = rp(K ′ /k).

Theorem 4.1. In any q-nite extension K/k, lm(rp(K/k)/k) is the only minimal intermediate eld of K/k over which K is uq-modular.

For the proof of this theorem we need the following result.