Degree Bounds for Splitting Fields (Separable Case)

Theorem (Separable bound).
Let $K$ be a field and let $f\in K[x]$ be a separable polynomial of degree $n$. If $L$ is the splitting field of $f$ over $K$, then $L/K$ is finite and

Equivalently, the Galois group injects into the permutation group on the $n$ roots, so its order is at most $n!$.

Examples.

1. Over $\mathbb{Q}$, $f=x^3-2$ is separable of degree $3$. Its splitting field has degree $6$, and indeed $6\le 3!=6$.
2. Over $\mathbb{Q}$, $f=x^4-2$ has a splitting field of degree $8$, and $8\le 4!=24$.
3. If $f$ already splits over $K$, then $L=K$ and $[L:K]=1\le n!$.

Related. splitting fields , separable ⇔ distinct roots .