Let $L/K$ be a Galois extension (I am interested in $\overline{\mathbb{Q}}/\mathbb{Q}$ so I do not assume it to be finite).

Let $V\subset L^n$ be a $L$-subvector space, of dimension $d$, such that $g(V)=V$ for each $g\in \mathrm{Gal}(L/K)$. Do you have a reference for the fact that the dimension of the $K$-vector space $V\cap K^n$ is equal to $d$?

It seems to me that it corresponds to a Galois descent and I've looked into books for this notion but I did not find something with this precise statement.