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1. CJM 1998 (vol 50 pp. 1007)

Elder, G. Griffith
Galois module structure of ambiguous ideals in biquadratic extensions
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
Categories:11R33, 11S15, 20C32

2. CJM 1997 (vol 49 pp. 722)

Elder, G. Griffith; Madan, Manohar L.
Galois module structure of the integers in wildly ramified $C_p\times C_p$ extensions
Let $L/K$ be a finite Galois extension of local fields which are finite extensions of $\bQ_p$, the field of $p$-adic numbers. Let $\Gal (L/K)=G$, and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$, respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We determine, explicitly in terms of specific indecomposable $\bZ_p[G]$-modules, the $\bZ_p[G]$-module structure of $\euO_L$ and $\euP_L$, for $L$, a composite of two arithmetically disjoint, ramified cyclic extensions of $K$, one of which is only weakly ramified in the sense of Erez \cite{erez}.

Keywords:Galois module structure---integral representation.
Categories:11S15, 20C32

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