Uniqueness of splitting fields

Statement

Let $K$ be a field, let $f \in K[x]$ be nonzero, and let $L$ and $L'$ be splitting fields of $f$ over $K$. Then there is a ring isomorphism $\sigma \colon L \to L'$ with $\sigma(c) = c$ for every $c \in K$.

Remarks

The proof is by induction on $[L : K]$. It extends an isomorphism one root at a time, with the structure of simple extensions. The isomorphism $\sigma$ is usually not unique. The isomorphisms from $L$ to itself that fix $K$ form the Galois group of $f$. So one speaks of "the" splitting field of $f$ over $K$. The choice of the irreducible factors in the construction does not change the result, up to isomorphism.

Explanations

L1 human

Two splitting fields of the same polynomial are the same field with different names for the elements. The proof extends a matching of the base field, one root at a time.