algebraically closed en · ADJ
Meanings
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(not-comparable, of a field, qualifier:field theory) Which contains as an element every root of every nonconstant univariate polynomial definable over it (i.e., over said field).
In many ways #92;Q#95;p is analogous to #92;R. For example, #92;R is not algebraically closed. The exercises below show that #92;Q#95;p is not algebraically closed. However, by adjoining i#61;#92;sqrt#123;-1#125; to #92;R, we get the field of complex numbers, which is algebraically closed. In contrast, the algebraic closure #92;overline#92;Q#95;p of #92;Q#95;p is not of finite degree over #92;Q. Moreover, #92;Complex is complete with respect to the extension of the usual norm of #92;R. Unfortunately, #92;overline#92;Q#95;p is not complete with respect to the extension of the p-adic norm. So after completing it (via the usual method of Cauchy sequences) we get a still larger field, usually denoted by #92;Complex#95;p, and it turns out to be both algebraically closed and complete.
2008, M. Ram Murty, Problems in Analytic Number Theory, 2nd edition, Springer, page 155:Definition 5.3.14. Let F be a field. A field extension P is called an algebraic closure of F if P is algebraically closed and every proper subfield of P containing F is not algebraically closed. In other words, the algebraic closure of F is the minimal algebraically closed field containing F.
2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 195: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:The fundamental theorem of algebra states that the field of complex numbers, #92;mathbb#123;C#125;, is algebraically closed.
- (not-comparable, of a group) Such that any finite set of equations and inequations has a solution.
Translations (7)
fr algébriquement clos (that contains among its elements every root of every nonconstant single-variable polynomial definable over it) · pl algebraicznie domknięty (that contains among its elements every root of every nonconstant single-variable polynomial definable over it) · pt algebricamente fechado (that contains among its elements every root of every nonconstant single-variable polynomial definable over it) · es algebraicamente cerrado (that contains among its elements every root of every nonconstant single-variable polynomial definable over it) · sv algebraiskt sluten (that contains among its elements every root of every nonconstant single-variable polynomial definable over it) · no algebraiskt lukket (that contains among its elements every root of every nonconstant single-variable polynomial definable over it) · it algebricamente chiuso (that contains among its elements every root of every nonconstant single-variable polynomial definable over it)