Parametric Galois extensions

Legrand, François

doi:10.1016/j.jalgebra.2014.07.034

Cited by 10 publications

(18 citation statements)

References 33 publications

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“…At the cost of some technical adjustments, similar criteria also hold for more general base fields. This is explained in [Leg15]. 4.2.2.…”

Section: Non Parametric Extensions Over Number Fieldsmentioning

confidence: 96%

“…The more general property for which the condition is required for any Galois extension F/k of group a given subgroup H ⊂ G is studied in [Leg15].…”

Section: Non Parametric Extensions Over Number Fieldsmentioning

confidence: 99%

“…We give here ( §4.3) -a result on four branch point Galois extensions (corollary 4.4); examples with G = Z/2Z and G = A 5 are then given (corollaries 4.5-6), -for each integer n ≥ 3, a practical result with G = S n (corollary 4.7). Many other examples are given in [Leg15], which is specifically devoted to parametric extensions, and in [Leg13, chapters 2-3].…”

Section: Introductionmentioning

confidence: 99%

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Specialization results and ramification conditions

Legrand

2016

Isr. J. Math.

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Abstract. Given a hilbertian field k of characteristic zero and a finite Galois extension E/k(T ) with group G such that E/k is regular, we produce some specializations of E/k(T ) at points t 0 ∈ P 1 (k) which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of Q of various finite groups with specified local behavior -ramified or unramified -at finitely many given primes. Secondly, in the case k is a number field, we provide criteria for the extension E/k(T ) to satisfy this property: at least one Galois extension F/k of group G is not a specialization of E/k(T ).

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“…At the cost of some technical adjustments, similar criteria also hold for more general base fields. This is explained in [Leg15]. 4.2.2.…”

Section: Non Parametric Extensions Over Number Fieldsmentioning

confidence: 96%

“…The more general property for which the condition is required for any Galois extension F/k of group a given subgroup H ⊂ G is studied in [Leg15].…”

Section: Non Parametric Extensions Over Number Fieldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Specialization results and ramification conditions

Legrand

2016

Isr. J. Math.

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“…Legrand gives the following criterion (see [5], Theorem 4.2): 1 Proposition 3.1. Let F 1 |K(t) and F 2 |K(t) be two regular Galois extensions with group G with ramification polynomials m 1 and m 2 .…”

Section: A Non-parametricity Criterionmentioning

confidence: 99%

“…One may further ask how many regular G-extensions are necessary to cover all G-extensions of K. This leads to the concept of G-parametric Galois extensions, introduced by Legrand in [5]. Definition 1.1 (G-parametric Galois extension).…”

Section: Introductionmentioning

confidence: 99%

Non-parametricity of rational translates of regular Galois extensions

König

2017

Acta Arith.

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We generalize a result of F. Legrand about the existence of non-parametric Galois extensions for a given group G. More precisely, for a K-regular Galois extension F |K(t), we consider the translates F (s)|K(s) by an extension K(s)|K(t) of rational function fields (in other words, s is a root of g(X) − t for some rational function g ∈ K(X)). We then show that if F |K(t) is a K-regular Galois extension with group G over a number field K, then for any degree k ≥ 2 and almost all (in a density sense) rational functions g of degree k, the translate of F by a root field of g(X)−t over K(t) is non-G-parametric, i.e. not all Galois extensions of K with group G arise as specializations of F (s)|K(s).

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Non-parametric sets of regular realizations over number fields

König

Legrand

2018

Journal of Algebra

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Given a number field k, we show that, for many finite groups G, all the Galois extensions of k with Galois group G cannot be obtained by specializing any given finitely many Galois extensions E/k(T ) with Galois group G and E/k regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions E/k(T ) with Galois group G and E/k regular of a certain type, under a conjectural "uniform Faltings' theorem".1 4 In a recent paper of Neftin and the two authors, a completely different method is used to show that A n (n ≥ 4) has no finite 1-parametric set over k; see [KLN17, Corollary 7.3]. We also mention this weaker result (which can be used with simple or non-simple groups): any given non-trivial regular Galois group G over k is the Galois group of a k-regular Galois extension of k(T ) that is not parametric over k; see [Leg16a, Theorem 1.3] and [Kön17, Theorem 2.2]. 5 i.e., if there exists no k-regular Galois extension of k(T ) with Galois group G. 6 Replace T − t 0 by 1/T if t 0 = ∞. 7 One has r = 0 if and only if G is trivial.

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Parametric Galois extensions

Cited by 10 publications

References 33 publications

Specialization results and ramification conditions

Specialization results and ramification conditions

Non-parametricity of rational translates of regular Galois extensions

Non-parametric sets of regular realizations over number fields

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