Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Self-Dual Integral Normal Bases and Galois Module Structure   Let $N/F$ be an odd degree Galois extension of number fields with Galois
group $G$ and rings of integers ${\mathfrak O}_N$ and ${\mathfrak O}_F=\bo$
respectively. Let $\mathcal{A}$ be the unique fractional ${\mathfrak
O}_N$-ideal with square equal to the inverse different of $N/F$. Erez has shown
that $\mathcal{A}$ is a locally free ${\mathfrak O}[G]$-module if and only if
$N/F$ is a so called weakly ramified extension. There have been a number of
results regarding the freeness of $\mathcal{A}$ as a $\Z[G]$-module, however
this question remains open. In this paper we prove that $\mathcal{A}$ is free
as a $\Z[G]$-module assuming that $N/F$ is weakly ramified and under the
hypothesis that for every prime $\wp$ of ${\mathfrak O}$ which ramifies wildly
in $N/F$, the decomposition group is abelian, the ramification group is cyclic
and $\wp$ is unramified in $F/\Q$.
  We make crucial use of a construction due to the first named author which
uses Dwork's exponential power series to describe self-dual integral normal
bases in Lubin-Tate extensions of local fields. This yields a new and striking
relationship between the local norm-resolvent and Galois Gauss sum involved.
Our results generalise work of the second named author concerning the case of
base field $\Q$.

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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: alg-geom
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The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas