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Let $X$ be $\mathbb{Z}$-module of finite cardinality $n$ and $\mu_n$ the group of $n$th roots of unity in $\bar{K}$. Suppose further that $X$ carries a $G_{K}$-operation (the absolute Galois group). Let $X^*$ be ${\rm Hom}_{\mathbb{Z}}(X,\mu_n)$ with the usual $G_K$ action $\sigma f(x)=\sigma(f(\sigma^{-1}x))$.

Then I want to understand the $G_K$-isomorphism $X\otimes X^*\cong \mu_n$. How can the canonical map $(x,x')\mapsto x'(x)$ be injective, in light of the usual linear algebra fact $V\otimes {\rm Hom}_F(V,F)\cong {\rm End}_F(V)$, for a finite dimensional $F$ vector space $V$. For a contradiction take $F=\mathbb{F}_p,V=F^2$: Then the dimensions of End and $\mu_n$ don't quite agree.

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