The Stacks project

Lemma 11.8.4. Consider a finite central skew field $K$ over $k$. Let $d^2 = [K : k]$. For any finite splitting field $k'$ for $K$ the degree $[k' : k]$ is divisible by $d$.

Proof. By Theorem 11.8.2 there exists a finite central simple algebra $B$ in the Brauer class of $K$ such that $[B : k] = [k' : k]^2$. By Lemma 11.5.1 we see that $B = \text{Mat}(n \times n, K)$ for some $n$. Then $[k' : k]^2 = n^2d^2$ whence the result. $\square$

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