= Galois descent of vector spaces
{c}
{title2=$L\otimes_KV^G\cong V$}
= Galois descent
{c}
{synonym}
For a <Finite Galois extension> $L/K$ and a finite-dimensional $L$-space $V$ with a <semilinear action> of its <Galois group>, the map $L\otimes_KV^G\to V$ is an isomorphism. The vectors $\sum_{\sigma}\sigma(a)\sigma(v)$ are invariant and span $V$ by the <Artin independence theorem>. Choosing an invariant $L$-basis then identifies the invariant space with its $K$-span. The proof does not divide by the extension degree.
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