algebraically closed
Word Rarity
Very rare
Frequency: 2.17 in modern English
30
Scrabble Score
19
Letters
5
Syllables
12
Unique Chars
Meaning
adj
Which contains as an element every root of every nonconstant univariate polynomial definable over it (i.e., over said field).
adj
Such that any finite set of equations and inequations has a solution.
Example Sentences
- "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."
- "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."
- "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."