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

Manjul Bhargava; Matthew Satriano

Displaying similar documents to “On a notion of “Galois closure” for extensions of rings”

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 $n \equiv 0 (m o d 8)$ , or $n \equiv 2 (m o d 8)$ 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 $Q$ .

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

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 $ℂ ((z))$ 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 $ℂ ((z))$ .

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...

Galois structure of ideals in wildly ramified abelian $p$ -extensions of a $p$ -adic field, and some applications

Nigel P. Byott (1997)

Journal de théorie des nombres de Bordeaux

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Let $K$ be a finite extension of $ℚ_{p}$ with ramification index $e$ , and let $L / K$ be a finite abelian $p$ -extension with Galois group $Γ$ and ramification index $p^{n}$ . We give a criterion in terms of the ramification numbers $t_{i}$ for a fractional ideal $𝔓^{h}$ of the valuation ring $S$ of $L$ not to be free over its associated order $𝔄 (K Γ; 𝔓^{h})$ . In particular, if $t_{n} - [t_{n} / p] < p^{n - 1} e$ then the inverse different can be free over its associated order only when $t_{i} \equiv - 1$ (mod $p^{n}$ ) for all $i$ . We give three consequences of this. Firstly, if $𝔄 (K Γ; S)$ is a Hopf order and...

The Sylow p-Subgroups of Tame Kernels in Dihedral Extensions of Number Fields

Qianqian Cui, Haiyan Zhou (2013)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let F/E be a Galois extension of number fields with Galois group $D_{2 ⁿ}$ . In this paper, we give some expressions for the order of the Sylow p-subgroups of tame kernels of F and some of its subfields containing E, where p is an odd prime. As applications, we give some results about the order of the Sylow p-subgroups when F/E is a Galois extension of number fields with Galois group $D_{16}$ .

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 $p \equiv 1 (m o d 2^{n - 1})$ , there exist unique real and complex...

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 $ℚ$ . 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 $ℚ^{(m^{'})} / K$ , where $m^{'}$ is the conductor...

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...

The Galois group of $X^{p} + a X^{s} + a$

B. Bensebaa, A. Movahhedi, A. Salinier (2008)

Acta Arithmetica

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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 $i \geq 2$ the étale wild kernels $W K_{2 i - 2}^{\overset{´}{e} t} (F)$ are defined as kernels of certain localization maps on the $i$ -fold twist of the $p$ -adic étale cohomology groups of $spec o_{F} [\frac{1}{p}]$ . 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...

Galois towers over non-prime finite fields

Alp Bassa, Peter Beelen, Arnaldo Garcia, Henning Stichtenoth (2014)

Acta Arithmetica

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We construct Galois towers with good asymptotic properties over any non-prime finite field $_{ℓ}$ ; that is, we construct sequences of function fields = (N₁ ⊂ N₂ ⊂ ⋯) over $_{ℓ}$ of increasing genus, such that all the extensions $N_{i} / N_{1}$ are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with...

An analogue of Pfister's local-global principle in the burnside ring

Martin Epkenhans (1999)

Journal de théorie des nombres de Bordeaux

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Let $N / K$ be a Galois extension with Galois group $𝒢$ . We study the set $𝒯 (𝒢)$ of $ℤ$ -linear combinations of characters in the Burnside ring $ℬ (𝒢)$ which give rise to $ℤ$ -linear combinations of trace forms of subextensions of $N / K$ which are trivial in the Witt ring W $(K)$ of $K$ . In particular, we prove that the torsion subgroup of $ℬ (𝒢) / 𝒯 (𝒢)$ coincides with the kernel of the total signature homomorphism.

On double covers of the generalized alternating group $ℤ_{d} ≀_{m}$ as Galois groups over algebraic number fields

Martin Epkenhans (1997)

Acta Arithmetica

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Let $ℤ_{d} ≀$ m $b e t h e g e n e r a l i z e d a l t e r n a t i n g g r o u p . W e p r o v e t h a t a l l d o u b l e c o v e r s o f$ ℤd ≀ m $o c c u r a s G a l o i s g r o u p s o v e r a n y a l g e b r a i c n u m b e r f i e l d . W e f u r t h e r r e a l i z e s o m e o f t h e s e d o u b l e c o v e r s a s t h e G a l o i s g r o u p s o f r e g u l a r e x t e n s i o n s o f ℚ (T) . I f d i s o d d a n d m > 7, t h e n e v e r y c e n t r a l e x t e n s i o n o f$ ℤd ≀ m $o c c u r s a s t h e G a l o i s g r o u p o f a r e g u l a r e x t e n s i o n o f ℚ (T) . W e f u r t h e r i m p r o v e s o m e o f o u r e a r l i e r r e s u l t s c o n c e r n i n g d o u b l e c o v e r s o f t h e g e n e r a l i z e d s y m m e t r i c g r o u p$ ℤd ≀ m $.$

Two classes of almost Galois coverings for algebras

Piotr Dowbor, Adam Hajduk (2012)

Colloquium Mathematicae

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We prove that for any representation-finite algebra A (in fact, finite locally bounded k-category), the universal covering F: Ã → A is either a Galois covering or an almost Galois covering of integral type, and F admits a degeneration to the standard Galois covering F̅: Ã→ Ã/G, where $G = Π (Γ_{A})$ is the fundamental group of $Γ_{A}$ . It is shown that the class of almost Galois coverings F: R → R’ of integral type, containing the series of examples from our earlier paper [Bol. Soc. Mat. Mexicana 17 (2011)],...

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, $b_{1} > 1 / 2 \cdot p e_{0} / (p - 1)$ where $e_{0}$ denotes the absolute ramification index of $K$ . Under this loose condition, we explicitly determine the $ℤ_{p} [G]$ -module structure of the ring of integers of $L$ , where $ℤ_{p}$ denotes the $p$ -adic integers and $G$ denotes the Galois group Gal $(L / K)$ . In the process of determining...

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 $1 \leq ρ \leq r_{2} + 1$ , ( $r_{2}$ 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.