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# Galois module structure of ambiguous ideals in biquadratic extensions

Published:1998-10-01
Printed: Oct 1998
• G. Griffith Elder
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## Abstract

Let $N/K$ be a biquadratic extension of algebraic number fields, and $G=\Gal (N/K)$. Under a weak restriction on the ramification filtration associated with each prime of $K$ above $2$, we explicitly describe the $\bZ[G]$-module structure of each ambiguous ideal of $N$. We find under this restriction that in the representation of each ambiguous ideal as a $\bZ[G]$-module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone. For a given group, $G$, define ${\cal S}_G$ to be the set of indecomposable $\bZ[G]$-modules, ${\cal M}$, such that there is an extension, $N/K$, for which $G\cong\Gal (N/K)$, and ${\cal M}$ is a $\bZ[G]$-module summand of an ambiguous ideal of $N$. Can ${\cal S}_G$ ever be infinite? In this paper we answer this question of Chinburg in the affirmative.
 Keywords: Galois module structure, wild ramification
 MSC Classifications: 11R33 - Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] 11S15 - Ramification and extension theory 20C32 - Representations of infinite symmetric groups