algebraically closed

Adjective

algebraically closed (not comparable)

1. (field theory) Which contains as an element every root of every nonconstant univariate polynomial definable over it (i.e., over said field).
   • The fundamental theorem of algebra states that the field of complex numbers, #92;mathbb#123;C#125;, is algebraically closed.
   • Definition. A field is algebraically closed when it satisfies the equivalent conditions in Proposition 4.1. #92;square
   • For instance, the fundamental theorem of algebra (Theorem III.8.11) states that #92;Complex is algebraically closed. The fields #92;R, #92;Q, #92;Z#95;p, are not algebraically closed, but #92;R and #92;Q can be embedded into the algebraically closed field #92;Complex.2007, Pierre Antoine Grillet, Abstract Algebra, 2nd edition, Springer, page 166
2. Such that any finite set of equations and inequations has a solution.