Meaning of algebraic closure | Babel Free

Definitions

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).

Equivalents

Examples

“The calculation of these various Galois groups leads to a determination of the algebraic closures of the ground fields in the splitting fields of the corresponding vectorial polynomials.”

“It turns out that the algebraic closure #92;mathbf#123;Q#125;#92;mathrm#123;a#125;#95;p is not complete, so we shall consider its completion #92;mathbf#123;C#125;#95;p: This field turns out to be algebraically closed and is a natural domain for the study of "analytic functions."”

“While #92;Complex contains an algebraic closure of #92;mathbb#123;Q#125;, it is by no means the only algebraically closed field containing an algebraic closure of #92;mathbb#123;Q#125;. We denote by #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; the algebraic closure of #92;mathbb#123;Q#125; in #92;Complex; this field is simply the subfield of #92;Complex consisting of algebraic numbers. The field #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; is isomorphic, then, to any algebraic closure of #92;mathbb#123;Q#125;, but even knowing that it is unique up to isomorphism very likely leaves us no more familiar with #92;mathbb#123;Q#125;#92;mathrm#123;alg#125; than we were.”

CEFR level