Uniqueness of Algebraic Closures - Imported Knowlpedia Content

Let $K$ be a field. An algebraic closure of $K$ is a field extension $\Omega/K$ that is algebraic and algebraically closed.

Theorem (uniqueness up to $K$ -isomorphism). If $\Omega$ and $\Omega'$ are algebraic closures of $K$ , then there exists a field isomorphism

\varphi:\Omega \xrightarrow{\ \sim\ } \Omega'

such that $\varphi|_K=\mathrm{id}_K$ . Equivalently, the algebraic closure of $K$ is unique up to (non-canonical) isomorphism over $K$ .

Remarks

A common way to phrase this is: once existence is known (see existence of algebraic closures), “the” algebraic closure $\overline K$ is determined uniquely up to $K$ -isomorphism.

Useful strengthening. Any $K$ -embedding $\Omega\hookrightarrow \Omega'$ is automatically surjective, hence an isomorphism: the image is an algebraically closed subfield of $\Omega'$ that is algebraic over $K$ , and an algebraic extension of an algebraically closed field must be trivial.

Examples

$K=\mathbb{Q}$ . The field of algebraic numbers $\overline{\mathbb{Q}}\subset \mathbb{C}$ is an algebraic closure of $\mathbb{Q}$ . If $\Omega$ is any other algebraic closure of $\mathbb{Q}$ , the theorem gives a $\mathbb{Q}$ -isomorphism $\Omega\cong \overline{\mathbb{Q}}$ , but there is no preferred (canonical) choice of such an isomorphism.

$K=\mathbb{R}$ . Since $\mathbb{C}=\mathbb{R}(i)$ is a simple extension generated by an algebraic element ( $i^2+1=0$ ), the extension $\mathbb{C}/\mathbb{R}$ is algebraic and $\mathbb{C}$ is algebraically closed; hence $\mathbb{C}$ is an algebraic closure of $\mathbb{R}$ . Uniqueness says any algebraic closure of $\mathbb{R}$ is $\mathbb{R}$ -isomorphic to $\mathbb{C}$ . The non-canonicity is visible in the nontrivial field automorphism of $\mathbb{C}$ over $\mathbb{R}$ , namely complex conjugation.

$K=\mathbb{F}_p$ . One can build an algebraic closure of $\mathbb{F}_p$ as a union $\bigcup_{n\ge1}\mathbb{F}_{p^n}$ inside a fixed ambient closure, where each $\mathbb{F}_{p^n}$ is a finite field (compare existence of finite fields). The theorem implies any two such “unions of finite fields” are isomorphic over $\mathbb{F}_p$ .