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Published online by Cambridge University Press: 20 November 2018
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.