Automorphism-count bound for a finite field extension
= Automorphism-count bound for a finite field extension
If $K/k$ is finite, then any distinct $k$-automorphisms $\sigma_1,\ldots,\sigma_m$ of $K$ are linearly independent in the $K$-vector space $\operatorname{Hom}_k(K,K)$, whose dimension over $K$ is $[K:k]$. Hence $m\leq[K:k]$. In particular, a finite automorphism group $G$ satisfies $|G|\leq[K:K^G]$.