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.



Galois descent for dimension of vector spaces
What you need is the corollary on the bottom of page 60 of Bourbaki, *Algèbre*, chapitre V : Corps commutatifs ; section 10 (Extensions galoisiennes), subsection 4 (Descente galoisienne).


The statement given there is more precise: given an $L$-subspace $W$ of $L^n$ which is stable under Galois, the $K$-subspace $V=W\cap K^n$ of $K^n$ satisfies $W=V\otimes\_K L$ and is the unique one with this property. Since dimension is preserved by base change, one has $\dim\_L(W)=\dim\_K(V)$, as you requested.


