Uniqueness of splitting fields

Let $K$ be a field and let $S\subseteq K[x]$ be a set of polynomials. Suppose $L$ and $L'$ are splitting fields of $S$ over $K$.

Theorem (uniqueness up to $K$-isomorphism). There exists a $K$-embedding $\varphi:L\to L'$ whose image is all of $L'$; in particular, $\varphi$ is an isomorphism of fields fixing $K$ pointwise. Thus splitting fields of the same data are unique up to $K$-isomorphism.

A common strengthening: if one fixes an algebraic closure $\overline K$ and realizes both $L$ and $L'$ as subfields of $\overline K$ generated by all roots of $S$, then $L=L'$ as subfields of $\overline K$. This is the “uniqueness inside a fixed closure” version (compare existence and uniqueness of splitting fields ).

Examples

1. Choice of square root does not matter.
   Over $K=\mathbb{Q}$, the polynomial $x^2-2$ has roots $\pm\sqrt2$. The splitting field generated by $\sqrt2$ is $\mathbb{Q}(\sqrt2)$, and the splitting field generated by $-\sqrt2$ is the same field. The $K$-automorphism $\sqrt2\mapsto -\sqrt2$ exhibits the uniqueness.

2. Cubic splitting fields.
   For $f=x^3-2\in\mathbb{Q}[x]$, any splitting field over $\mathbb{Q}$ is $K$-isomorphic to $\mathbb{Q}(\sqrt[3]{2},\zeta_3)$, even if one starts by adjoining a different real cube root or different primitive cube root of unity. This matches the fact that a normal extension is characterized as a splitting field (see normality and splitting fields ).

3. Finite fields.
   Any two fields of size $p^n$ are isomorphic (see existence and uniqueness of finite fields ). In particular, if $L$ is the splitting field over $\mathbb{F}_p$ of an irreducible degree-$n$ polynomial, then $L\cong \mathbb{F}_{p^n}$ regardless of which irreducible polynomial one starts with.