algebraic closure

Noun

 * 1)  A field G such that every polynomial over F splits completely over G (i.e., every element of G is a root of some polynomial over F and every root of every polynomial over F is an element of G).

Usage notes

 * Notations for the algebraic closure of a field $$F$$ include $$\overline{F}$$ and $$F^\mathrm{a}$$.
 * Using Zorn's lemma (or the weaker ), it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Consequently, authors often speak of the (rather than an) algebraic closure of K. (See )
 * The field of complex numbers, $$\Complex$$, is the algebraic closure of the field of real numbers, $$\R$$.
 * The algebraic closure of the field of p-adic numbers, $$\mathbb{Q}_p$$, is denoted $$\overline\mathbb{Q}_p$$ or $$\mathbb{Q}^\mathrm{a}_p$$. (Unlike $$\Complex$$, and indeed unlike $$\mathbb{Q}_p$$, $$\overline\mathbb{Q}_p$$ is not metrically complete: its metric completion, which is algebraically closed, is denoted $$\Complex_p$$ or $$\Omega_p$$.)

Translations

 * Finnish: algebrallinen sulkeuma
 * Portuguese: fecho algébrico