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Realizable Galois module classes over the group ring for non abelian extensions

Nigel P. Byott, Bouchaïb Sodaïgui (2013)

Annales de l’institut Fourier

Given an algebraic number field $k$ and a finite group $Γ$ , we write $ℛ (O_{k} [Γ])$ for the subset of the locally free classgroup $Cl (O_{k} [Γ])$ consisting of the classes of rings of integers $O_{N}$ in tame Galois extensions $N / k$ with $Gal (N / k) ≅ Γ$ . We determine $ℛ (O_{k} [Γ])$ , and show it is a subgroup of $Cl (O_{k} [Γ])$ by means of a description using a Stickelberger ideal and properties of some cyclic codes, when $k$ contains a root of unity of prime order $p$ and $Γ = V ⋊ C$ , where $V$ is an elementary abelian group of order $p^{r}$ and $C$ is a cyclic group of order $m > 1$ acting faithfully on...

Relative Galois module structure of integers of abelian fields

Nigel P. Byott, Günter Lettl (1996)

Journal de théorie des nombres de Bordeaux

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 of $K$ .

Relative Galois module structure of rings of integers of cyclomatic fields.

Chong-Hai Lim, Shih-Ping Chan (1993)

Journal für die reine und angewandte Mathematik

Relative integral basis for algebraic number fields.

Haghighi, Mahmood (1986)

International Journal of Mathematics and Mathematical Sciences

Remarks on normal bases

Marcin Mazur (2001)

Colloquium Mathematicae

We prove that any Galois extension of a commutative ring with a normal basis and abelian Galois group of odd order has a self-dual normal basis. We apply this result to get a very simple proof of nonexistence of normal bases for certain extensions which are of interest in number theory.

Résultats récents sur la monogénéité de certains anneaux d'entiers

J. COUGNARD (1987/1988)

Seminaire de Théorie des Nombres de Bordeaux

Rings of Integers and Trace Forms for Tame Extensions of Odd Degree.

M.J. Taylor (1989)

Mathematische Zeitschrift

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