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algebraically closed
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algebraically closed (not comparable)
- (algebra, field theory, of a field) 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, , is algebraically closed.
- 2007, Pierre Antoine Grillet, Abstract Algebra, Springer, 2nd Edition, page 166,
- Definition. A field is algebraically closed when it satisfies the equivalent conditions in Proposition 4.1.
- For instance, the fundamental theorem of algebra (Theorem III.8.11) states that is algebraically closed. The fields , , , are not algebraically closed, but and can be embedded into the algebraically closed field .
- 2008, M. Ram Murty, Problems in Analytic Number Theory, Springer, 2nd Edition, page 155,
- In many ways is analogous to . For example, is not algebraically closed. The exercises below show that is not algebraically closed. However, by adjoining to , we get the field of complex numbers, which is algebraically closed. In contrast, the algebraic closure of is not of finite degree over . Moreover, is complete with respect to the extension of the usual norm of . Unfortunately, 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 , and it turns out to be both algebraically closed and complete.
- 2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 195,
- Definition 5.3.14. Let be a field. A field extension is called an algebraic closure of if is algebraically closed and every proper subfield of containing is not algebraically closed.
- In other words, the algebraic closure of is the minimal algebraically closed field containing .
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