Galois finite extension Let $K/ \mathbb{Q}$ be a finite Galois extension, $K \otimes_{\mathbb{Q}} \mathbb{R} \simeq \mathbb{R}^s \oplus \mathbb{C}^t$. 
Prove that either $s=0$ or $t=0$. 

The automorphisms of $K/\Bbb Q$ are continuous, so they extend to a completion. If we denote an infinite place by $\mathfrak{p}$, we know one completion is $K_{\mathfrak{p}}$. As all other completions are given by $\sigma(K)_{\sigma(\mathfrak{p})}$ and $\sigma(K)=K$ (since we are dealing with a Galois extension) we use the fact that $\text{Gal}\left(K/\Bbb Q\right)$ permutes the places transitively to conclude all completions are isomorphic as $\Bbb R$ vector spaces, in particular they have the same dimension.
Now as
$$K\otimes_{\Bbb R}\Bbb R \cong\prod_{\mathfrak{p}\text{ real}}K_{\mathfrak{p}}\times\prod_{\mathfrak{p}\text{ complex}}K_{\mathfrak{p}}$$
we conclude the result.
