Displaying similar documents to “Galois structure of ideals in wildly ramified abelian p -extensions of a p -adic field, and some applications”

Relative Galois module structure of integers of abelian fields

Nigel P. Byott, Günter Lettl (1996)

Journal de théorie des nombres de Bordeaux

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Let L / K be an extension of algebraic number fields, where L is abelian over . In this paper we give an explicit description of the associated order 𝒜 L / K of this extension when K is a cyclotomic field, and prove that o L , the ring of integers of L , is then isomorphic to 𝒜 L / K . This generalizes previous results of Leopoldt, Chan Lim and Bley. Furthermore we show that 𝒜 L / K is the maximal order if L / K is a cyclic and totally wildly ramified extension which is linearly disjoint to ( m ' ) / K , where m ' is the conductor...

Polynomials over Q solving an embedding problem

Nuria Vila (1985)

Annales de l'institut Fourier

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The fields defined by the polynomials constructed in E. Nart and the author in J. Number Theory 16, (1983), 6–13, Th. 2.1, with absolute Galois group the alternating group A n , can be embedded in any central extension of A n if and only if n 0 ( m o d 8 ) , or n 2 ( m o d 8 ) and n is a sum of two squares. Consequently, for theses values of n , every central extension of A n occurs as a Galois group over Q .

On a notion of “Galois closure” for extensions of rings

Manjul Bhargava, Matthew Satriano (2014)

Journal of the European Mathematical Society

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We introduce a notion of “Galois closure” for extensions of rings. We show that the notion agrees with the usual notion of Galois closure in the case of an S n degree n extension of fields. Moreover, we prove a number of properties of this construction; for example, we show that it is functorial and respects base change. We also investigate the behavior of this Galois closure construction for various natural classes of ring extensions.

Some remarks on Hilbert-Speiser and Leopoldt fields of given type

James E. Carter (2007)

Colloquium Mathematicae

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Let p be a rational prime, G a group of order p, and K a number field containing a primitive pth root of unity. We show that every tamely ramified Galois extension of K with Galois group isomorphic to G has a normal integral basis if and only if for every Galois extension L/K with Galois group isomorphic to G, the ring of integers O L in L is free as a module over the associated order L / K . We also give examples, some of which show that this result can still hold without the assumption that...

On Galois structure of the integers in cyclic extensions of local number fields

G. Griffith Elder (2002)

Journal de théorie des nombres de Bordeaux

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Let p be a rational prime, K be a finite extension of the field of p -adic numbers, and let L / K be a totally ramified cyclic extension of degree p n . Restrict the first ramification number of L / K to about half of its possible values, b 1 > 1 / 2 · p e 0 / ( p - 1 ) where e 0 denotes the absolute ramification index of K . Under this loose condition, we explicitly determine the p [ G ] -module structure of the ring of integers of L , where p denotes the p -adic integers and G denotes the Galois group Gal ( L / K ) . In the process of determining...

A valuation criterion for normal basis generators of Hopf-Galois extensions in characteristic p

Nigel P. Byott (2011)

Journal de Théorie des Nombres de Bordeaux

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Let S / R be a finite extension of discrete valuation rings of characteristic p > 0 , and suppose that the corresponding extension L / K of fields of fractions is separable and is H -Galois for some K -Hopf algebra H . Let 𝔻 S / R be the different of S / R . We show that if S / R is totally ramified and its degree n is a power of p , then any element ρ of L with v L ( ρ ) - v L ( 𝔻 S / R ) - 1 ( mod n ) generates L as an H -module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. G. Elder for Galois...

Galois co-descent for étale wild kernels and capitulation

Manfred Kolster, Abbas Movahhedi (2000)

Annales de l'institut Fourier

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Let F be a number field with ring of integers o F . For a fixed prime number p and i 2 the étale wild kernels W K 2 i - 2 e ´ t ( F ) are defined as kernels of certain localization maps on the i -fold twist of the p -adic étale cohomology groups of spec o F [ 1 p ] . These groups are finite and coincide for i = 2 with the p -part of the classical wild kernel W K 2 ( F ) . They play a role similar to the p -part of the p -class group of F . For class groups, Galois co-descent in a cyclic extension L / F is described by the ambiguous class formula given...

Differential Galois Theory for an Exponential Extension of ( ( z ) )

Magali Bouffet (2003)

Bulletin de la Société Mathématique de France

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In this paper we study the formal differential Galois group of linear differential equations with coefficients in an extension of ( ( z ) ) by an exponential of integral. We use results of factorization of differential operators with coefficients in such a field to give explicit generators of the Galois group. We show that we have very similar results to the case of ( ( z ) ) .

A note on free pro- p -extensions of algebraic number fields

Masakazu Yamagishi (1993)

Journal de théorie des nombres de Bordeaux

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For an algebraic number field k and a prime p , define the number ρ to be the maximal number d such that there exists a Galois extension of k whose Galois group is a free pro- p -group of rank d . The Leopoldt conjecture implies 1 ρ r 2 + 1 , ( r 2 denotes the number of complex places of k ). Some examples of k and p with ρ = r 2 + 1 have been known so far. In this note, the invariant ρ is studied, and among other things some examples with ρ < r 2 + 1 are given.

Quaternion extensions with restricted ramification

Peter Schmid (2014)

Acta Arithmetica

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In any normal number field having Q₈, the quaternion group of order 8, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes 2 and 3 are ramified. In this note we describe in detail all Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime p 1 ( m o d 2 n - 1 ) , there exist unique real and complex...

Galois extensions of height-one commuting dynamical systems

Ghassan Sarkis, Joel Specter (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider a dynamical system consisting of a pair of commuting power series under composition, one noninvertible and another nontorsion invertible, of height one with coefficients in the p -adic integers. Assuming that each point of the dynamical system generates a Galois extension over the base field, we show that these extensions are in fact abelian, and, using results from the theory of the field of norms, we also show that the dynamical system must include a torsion series. From...

Associated orders of certain extensions arising from Lubin-Tate formal groups

Nigel P. Byott (1997)

Journal de théorie des nombres de Bordeaux

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Let k be a finite extension of p , let k 1 , respectively k 3 , be the division fields of level 1 , respectively 3 , arising from a Lubin-Tate formal group over k , and let Γ = Gal( k 3 / k 1 ). It is known that the valuation ring k 3 cannot be free over its associated order 𝔄 in K Γ unless k = p . We determine explicitly under the hypothesis that the absolute ramification index of k is sufficiently large.