Uniqueness of Splitting Fields

Theorem (Uniqueness).
Let $f\in K[x]$ be a nonzero polynomial and let $L$ and $L'$ be splitting fields of $f$ over $K$. Then there exists a $K$-isomorphism

(i.e., an isomorphism fixing $K$ pointwise). The isomorphism is generally not unique.

This is the uniqueness part of splitting field existence/uniqueness .

Examples.

1. If $L=\mathbb{Q}(\sqrt2)$ and $L'=\mathbb{Q}(\alpha)$ where $\alpha^2=2$, then $\alpha=\pm\sqrt2$ and the map $\sqrt2\mapsto \alpha$ gives a $\mathbb{Q}$-isomorphism $L\cong_{\mathbb{Q}} L'$.
2. For $f=x^2+x+1$ over $\mathbb{F}_2$, any two splitting fields are isomorphic to $\mathbb{F}_4$.
3. For separable irreducible $f$, the splitting field is characterized (up to $K$-isomorphism) by “adjoin all roots.”

Related. field isomorphisms , splitting fields and normality .