algebraic closure

plural algebraic closures

1. field theoryalgebraA 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).

   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.

   1999, Shreeram S. Abhyankar, “Galois Theory of Semilinear Transformations”, in Helmut Voelklein, David Harbater, J. G. Thompson, Peter Müller, editors, Aspects of Galois Theory, Cambridge University Press, page 1

   2 more examples

   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."

   2000, Alain M. Robert, A Course in p-adic Analysis, Springer, page 127

   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.

   2004, John Swallow, Exploratory Galois Theory, Cambridge University Press, page 179