Galois module structure of the rings of integers in wildly ramified extensions

Stephen M. J. Wilson

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Displaying similar documents to “Galois module structure of the rings of integers in wildly ramified extensions”

Galois module structure of rings of integers

Martin J. Taylor (1980)

Annales de l'institut Fourier

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Let $E / F$ be a Galois extension of number fields with $Γ =$ Gal $(E / F)$ and with property that the divisors of $(E : F)$ are non-ramified in $E / Q$ . We denote the ring of integers of $E$ by $𝒪_{E}$ and we study $𝒪_{E}$ as a $Z Γ$ -module. In particular we show that the fourth power of the (locally free) class of $𝒪_{E}$ is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of $ℰ_{E}$ , together with new determinantal congruences for cyclic group rings and corresponding...

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 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...

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...

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...

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 $Q$ .

On the Galois structure of the square root of the codifferent

D. Burns (1991)

Journal de théorie des nombres de Bordeaux

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Let $L$ be a finite abelian extension of $ℚ$ , with $𝒪_{L}$ the ring of algebraic integers of $L$ . We investigate the Galois structure of the unique fractional $𝒪_{L}$ -ideal which (if it exists) is unimodular with respect to the trace form of $L / ℚ$ .