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Publication# Construction Of Self-Dual Integral Normal Bases In Abelian Extensions Of Finite And Local Fields

Résumé

Let $F/E$ be a finite Galois extension of fields with abelian Galois group $\Gamma$. A self-dual normal basis for $F/E$ is a normal basis with the additional property that $Tr_{F/E}(g(x),h(x))=\delta_{g,h}$ for $g,h\in\Gamma$. Bayer-Fluckiger and Lenstra have shown that when $char(E)\neq 2$, then $F$ admits a self-dual normal basis if and only if $[F:E]$ is odd. If $F/E$ is an extension of finite fields and $char(E)=2$, then $F$ admits a self-dual normal basis if and only if the exponent of $\Gamma$ is not divisible by $4$. In this paper we construct self-dual normal basis generators for finite extensions of finite fields whenever they exist. Now let $K$ be a finite extension of $\Q_p$, let $L/K$ be a finite abelian Galois extension of odd degree and let $\bo_L$ be the valuation ring of $L$. We define $A_{L/K}$ to be the unique fractional $\bo_L$-ideal with square equal to the inverse different of $L/K$. It is known that a self-dual integral normal basis exists for $A_{L/K}$ if and only if $L/K$ is weakly ramified. Assuming $p\neq 2$, we construct such bases whenever they exist.

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If L/K is a finite Galois extension of local fields, then we say that the valuation criterion VC(L/K) holds if there is an integer d such that every element x is an element of L with valuation d generates a normal basis for L/K. Answering a question of Byott and Elder, we first prove that VC(L/K) holds if and only if the tamely ramified part of the extension L/K is trivial and every non-zero K[G]-submodule of L contains a unit of the valuation ring of L. Moreover, the integer d can take one value modulo [L:K] only, namely-d(L/K)-1, where d(L/K) is the valuation of the different of L/K. When K has positive characteristic, we thus recover a recent result of Elder and Thomas, proving that VC(L/K) is valid for all extensions L/K in this context. When char K=0, we identify all abelian extensions L/K for which VC(L/K) is true, using algebraic arguments. These extensions are determined by the behaviour of their cyclic Kummer subextensions.

2012This thesis is concerned with the algebraic theory of hermitian forms. It is organized in two parts. The first, consisting of the first two chapters, deals with some descent properties of unimodular hermitian forms over central simple algebras with involution. The second, which consists of the last two chapters, generalizes several classical properties of unimodular hermitian forms over rings with involution to the setting of sesquilinear forms in hermitian categories. The original results established in this thesis are joint work with Professor Eva Bayer-Fluckiger. The first chapter contains an introduction to the algebraic theory of unimodular ε-hermitian forms over fields with involution. One knows that if L/K is an extension of odd degree (where char(K) ≠ 2) then the restriction map rL/K : W(K) →W(L) is injective. In addition, if the extension is purely inseparable then the map rL/K is bijective. In the second chapter we first introduce the basic notions and techniques of the theory of unimodular ε-hermitian forms over algebras with involution, in particular the technique of Morita equivalence. Let L/K be a finite field extension, τ an involution on L and A a finite-dimensional K-algebra endowed with an involution α such that αœK = τœK. E. Bayer-Fluckiger and H.W. Lenstra proved that if L/K is of odd degree and αœK = idK then the restriction map rL/Kε : Wε(A, α) → Wε(A ⊗K L, α ⊗ τ) is injective for any ε = ±1. This holds also if αœK ≠ idK. We prove that if, in addition, L/K is purely inseparable and A is a central simple K-algebra, then the above map is actually bijective. The proof proceeds via induction on the degree of the algebra and uses in an essential way an exact sequence of Witt groups due to R. Parimala, R. Sridharan and V. Suresh, later extended by N. Gernier-Boley and M.G. Mahmoudi. The third chapter contains a survey of the theory of hermitian and quadratic forms in hermitian categories. In particular, we cover the transfer between two hermitian categories, the reduction by an ideal, the transfer into the endomorphism ring of an object, as well as the Krull-Schmidt-Azumaya theorem and some of its applications. In the fourth chapter we prove, adapting the ideas developed by E. Bayer-Fluckiger and L. Fainsilber, that the category of sesquilinear forms in a hermitian category ℳ is equivalent to the category of unimodular hermitian forms in the category of double arrows of ℳ. In order to obtain this equivalence of categories we associate to a sesquilinear form the double arrow consisting of its two adjoints, equipped with a canonical unimodular hermitian form. This equivalence of categories allows us to define a notion of Witt group for sesquilinear forms in hermitian categories. This generalizes the classical notion of a Witt group of unimodular hermitian forms over rings with involution. Using the above equivalence of categories we deduce analogues of the Witt cancellation theorem and Springer's theorem for sesquilinear forms over certain algebras with involution. We also extend some finiteness results due to E. Bayer-Fluckiger, C. Kearton and S.M. J. Wilson. In addition, we study the weak Hasse-Minkowski principle for sesquilinear forms over skew fields with involution over global fields. We prove that this principle holds for systems of sesquilinear forms over a skew field over a global field and endowed with a unitary involution. Two systems of sesquilinear forms are hence isometric if and only if they are isometric over all the completions of the base field. This result has already been known for unimodular hermitian and skew-hermitian forms over rings with involution, under the same hypothesis. Finally, we study the behaviour of the Witt group of a linear hermitian category under extension of scalars. Let K be a field of characteristic different from 2, L a finite extension of K and ℳ a K-linear hermitian category. We define the extension of ℳ to L as being the category with the same objects as ℳ and with morphisms given by the morphisms of ℳ extended to L. We obtain an L-linear hermitian category, denoted by ℳL. The canonical functor of scalar extension ℛL/K : ℳ → ℳL induces for any ε = ±1 a group homomorphism Wε(ℳ) →Wε(ℳL). We prove that if all the idempotents of the category ℳ split and the extension L/K is of odd degree then this homomorphism is injective. This result has already been known in the case when ℳ is the category of finite-dimensional K-vector spaces.

Let K be a finite extension of Q(p), let L/K be a finite abelian Galois extension of odd degree and let D-L be the valuation ring of L. We define A(L/K) to be the unique fractional D-L-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Q(p) contained in certain cyclotomic extensions, Erez has described integral normal bases for A(L)/Q(p) that are self-dual with respect to the trace form. Assuming K/Q(p) to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions. (C) 2009 Elsevier Inc. All rights reserved.

2009