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Let $A$ be a $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$-module which is a finitely generated free $\mathbb{Z}$-module. I'm interested in the behaviour of cohomology classes in $$\mathrm{H}^1(\mathbb …

Let $k$ be a local field with absolute Galois group $\Gamma_k$, inertia subgroup $I_k \subset \Gamma_k$, and residue field $\mathbb{F}$. Let $M$ be a finite Galois module over $k$. The unramified Galo …

I'll upgrade my comment to an answer. Any finite Galois extension of $\mathbb{Q}_l$ of degree coprime to $l$ is tamely ramified. In particular, its Galois group is an extension of two cyclic groups. …

Let $G$ be a semisimple algebraic group over $\mathbb{Q}_p$. Then by definition $G$ admits no non-trivial algebraic characters, i.e. homomorphisms $G \to \mathbb{G}_m$. However, it is quite possible …

As you already seem to know, unitary groups are classified by separable quadratic field extensions (in fact one should really work with separable quadratic algebras, i.e. also $F\times F$, correspondi …