Extensions régulières de Q(T) de groupe de galois Ãn

Mestre, Jean-François

doi:10.1016/0021-8693(90)90189-u

Cited by 56 publications

(44 citation statements)

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“…For each index i ∈ {1, 2, 3}, pick t i ∈ O i . From remark 3.11 and our assumption on the branch points, the polynomials m t 1 (T ) and (b) Let k be a number field such that Q(i) ⊂ k. From [Mes90], the splitting field over k(T ) of P (T, X) = (X 5 −X)−T (25X 4 −9) provides a Galois extension E/k(T ) of group A 5 with E/k regular. Its branch points are the roots of the polynomial S(T ) = 1 + (5 5 · 3 3 ) T 4 .…”

Section: Non Parametric Extensions Over Number Fieldsmentioning

confidence: 97%

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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Section: Non Parametric Extensions Over Number Fieldsmentioning

confidence: 97%

Specialization results and ramification conditions

Legrand

2016

Isr. J. Math.

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“…Form the polynomial H = ¡sA(S)res(P, R), where Is is the coefficient of the highest term of S, res(P, R) is the resultant of P and R, and A(S) is the discriminant of 5. We then have Theorem 2 [2]. Let p(X) be a specialization of P into Q such that H does not vanish, and let q(X) be the corresponding specialization of Q.…”

Section: Resultsmentioning

confidence: 98%

On constructing fields corresponding to the 𝐴̃_{𝑛}’s of Mestre for odd 𝑛

Swallow¹

1994

Proc. Amer. Math. Soc.

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“…So, Schinzel's original questions about reducibility of variables separated polynomials inspired methods influencing the literature around the Inverse Galois problem and ranks of abelian variety points over Q. For these topics we revisit the territory (and relationship of) [Fri90] and [Mes90]. The examples in this paper stress formulas connecting such families to covers of the classical j-line.…”

Section: 2mentioning

confidence: 99%

Variables separated polynomials, the genus 0 problem and moduli spaces

Fried¹

1999

Number Theory in Progress

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Abstract. The monodromy method-featuring braid group action-first appeared as a moduli space approach for finding solutions of arithmetic problems that produce reducible variables separated curves. Examples in this paper illustrate its most interesting aspect: investigating the moduli space of exceptions to a specific diophantine outcome. Explicit versions of Hilbert's irreducibility theorem and Davenport's problem fostered this technique and motivated the genus 0 problem started by J. Thompson and taken up by many group theorists. We review progress on the genus 0 problem in 0 characteristic, and its quite different contributions in positive characteristic.Example: Let f and h be polynomials with coefficients in a number field K. The classification of finite simple groups shows there is a bound on exceptional degrees for f to the following result. If f is indecomposable and h is not a composition with f , then f (x) − h(y) is irreducible. This answered challenge problems on factorization of variables separated polynomials posed by A. Schinzel in the early 60's. This limitation result holds, however, only in characteristic 0, one difference between the Genus 0 Problem here and in positive characteristic. Finite field example-Davenport's Problem: For each finite field Fq there are infinitely many surprising polynomial pairs (f, h) (of degree prime to the characteristic) whose value sets are equal over F q t , t = 1, 2, . . . . Though we include unpublished results from 30 years ago, a new set of problems stretch the methods. Example of a general theme: Let M d be the elements ofQ of degree no more than d over Q.Similarly, there is a value set V f (d) (degree 1 fiber). Following a reduction due to G. Frey, for any integer d, there are polynomials f of unbounded degreeThe full monodromy method finds precise arithmetic information by extending observations from modular curves. A semi-classical observation: Divisors with support in cusp points on modular curves generate a torsion group on the Jacobian of the curve. Illustrations here (from alternating group covers) show generalizations of this issue are ubiquitous with the monodromy method.

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Extensions régulières de Q(T) de groupe de galois Ãn

Cited by 56 publications

References 4 publications

Specialization results and ramification conditions

Specialization results and ramification conditions

On constructing fields corresponding to the 𝐴̃_{𝑛}’s of Mestre for odd 𝑛

Variables separated polynomials, the genus 0 problem and moduli spaces

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