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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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One dimensional (phi,Gamma)-modules in char p

The correspondence requires the $(\phi,\Gamma)$-module to have the \'etale property for its underlying $\phi$-module, and this plays an essential role in the proof of the correspondence (see Fontaine' …
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