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Galois Module Structure of Ambiguous Ideals in Biquadratic Extensions

  • G. Griffith Elder (a1)

Abstract

Let $N/K$ be a biquadratic extension of algebraic number fields, and $G\,=\,\text{Gal(}N/K\text{)}$ . Under a weak restriction on the ramification filtration associated with each prime of $K$ above 2, we explicitly describe the $\mathbb{Z}\text{ }[G]\text{ }$ -module structure of each ambiguous ideal of $N$ . We find under this restriction that in the representation of each ambiguous ideal as a $\mathbb{Z}\text{ }[G]\text{ }$ -module, the exponent (or multiplicity) of each indecomposable module is determined by the invariants of ramification, alone.

For a given group, $G$ , define ${{S}_{G}}$ to be the set of indecomposable $\mathbb{Z}\text{ }[G]\text{ }$ -modules, $M$ , such that there is an extension, $N/K$ , for which $G\cong \text{Gal(}N/K\text{)}$ , and $M$ is a $\mathbb{Z}\text{ }[G]\text{ }$ -module summand of an ambiguous ideal of $N$ . Can ${{S}_{G}}$ ever be infinite? In this paper we answer this question of Chinburg in the affirmative.

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Galois Module Structure of Ambiguous Ideals in Biquadratic Extensions

  • G. Griffith Elder (a1)

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