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The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.

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1-dimensional semi-stable Galois representations with coefficients

For any p-adic local field K, all 1-dim semi-stable Galois repn: $G_K \to Q_p^{*}$ are just $Q_p(n)\otimes \mu$, where $Q_p(n)$ is the Tate twist of cyclotomic character, and $\mu$ an unramified chara …