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 $t=0$. 

[Addendum] It is known that here $s$ is the number of distinct real embeddings of $K$, and $d$ is the number of pairs of complex conjugate pairs of embeddings of $K$. I set out to prove this form of the claim. IOW I seek to prove that either all the embeddings of $K$ are real or none is. [/Addendum]
As $\Bbb{C}$ is algebraically closed we can assume that $K$ is a subfield of $\Bbb{C}$. Let $[K:\Bbb{Q}]=n$. It is known that there are exactly $n$ embeddings $\sigma_1,\sigma_2,\ldots,\sigma_n$ from $K$ to $\Bbb{C}$. But we also know that $K$, being Galois, has $n$ distinct automorphisms. It follows that the above embeddings are exactly the automorphisms of $K$, and thus have the property $\sigma_i(K)=K$. As $K$ either is or is not a subset of the reals the claim follows from this.
