Examples
As an example, the field ofEquivalent properties
Given a field ''F'', the assertion "''F'' is algebraically closed" is equivalent to other assertions:The only irreducible polynomials are those of degree one
The field ''F'' is algebraically closed if and only if the only irreducible polynomials in theEvery polynomial is a product of first degree polynomials
The field ''F'' is algebraically closed if and only if every polynomial ''p''(''x'') of degree ''n'' ≥ 1, with coefficients in ''F'', splits into linear factors. In other words, there are elements ''k'', ''x''1, ''x''2, ..., ''xn'' of the field ''F'' such that ''p''(''x'') = ''k''(''x'' − ''x''1)(''x'' − ''x''2) ⋯ (''x'' − ''xn''). If ''F'' has this property, then clearly every non-constant polynomial in ''F'' 'x''has some root in ''F''; in other words, ''F'' is algebraically closed. On the other hand, that the property stated here holds for ''F'' if ''F'' is algebraically closed follows from the previous property together with the fact that, for any field ''K'', any polynomial in ''K'' 'x''can be written as a product of irreducible polynomials.Polynomials of prime degree have roots
If every polynomial over ''F'' of prime degree has a root in ''F'', then every non-constant polynomial has a root in ''F''. It follows that a field is algebraically closed if and only if every polynomial over ''F'' of prime degree has a root in ''F''.The field has no proper algebraic extension
The field ''F'' is algebraically closed if and only if it has no proper algebraic extension. If ''F'' has no proper algebraic extension, let ''p''(''x'') be some irreducible polynomial in ''F'' 'x'' Then the quotient of ''F'' 'x''modulo theThe field has no proper finite extension
The field ''F'' is algebraically closed if and only if it has no proper finite extension because if, within the previous proof, the term "algebraic extension" is replaced by the term "finite extension", then the proof is still valid. (Note that finite extensions are necessarily algebraic.)Every endomorphism of ''Fn'' has some eigenvector
The field ''F'' is algebraically closed if and only if, for each natural number ''n'', everyDecomposition of rational expressions
The field ''F'' is algebraically closed if and only if every rational function in one variable ''x'', with coefficients in ''F'', can be written as the sum of a polynomial function with rational functions of the form ''a''/(''x'' − ''b'')''n'', where ''n'' is a natural number, and ''a'' and ''b'' are elements of ''F''. If ''F'' is algebraically closed then, since the irreducible polynomials in ''F'' 'x''are all of degree 1, the property stated above holds by the theorem on partial fraction decomposition. On the other hand, suppose that the property stated above holds for the field ''F''. Let ''p''(''x'') be an irreducible element in ''F'' 'x'' Then the rational function 1/''p'' can be written as the sum of a polynomial function ''q'' with rational functions of the form ''a''/(''x'' – ''b'')''n''. Therefore, the rational expression : can be written as a quotient of two polynomials in which the denominator is a product of first degree polynomials. Since ''p''(''x'') is irreducible, it must divide this product and, therefore, it must also be a first degree polynomial.Relatively prime polynomials and roots
For any field ''F'', if two polynomials ''p''(''x''),''q''(''x'') ∈ ''F'' 'x''are relatively prime then they do not have a common root, for if ''a'' ∈ ''F'' was a common root, then ''p''(''x'') and ''q''(''x'') would both be multiples of ''x'' − ''a'' and therefore they would not be relatively prime. The fields for which the reverse implication holds (that is, the fields such that whenever two polynomials have no common root then they are relatively prime) are precisely the algebraically closed fields. If the field ''F'' is algebraically closed, let ''p''(''x'') and ''q''(''x'') be two polynomials which are not relatively prime and let ''r''(''x'') be their greatest common divisor. Then, since ''r''(''x'') is not constant, it will have some root ''a'', which will be then a common root of ''p''(''x'') and ''q''(''x''). If ''F'' is not algebraically closed, let ''p''(''x'') be a polynomial whose degree is at least 1 without roots. Then ''p''(''x'') and ''p''(''x'') are not relatively prime, but they have no common roots (since none of them has roots).Other properties
If ''F'' is an algebraically closed field and ''n'' is a natural number, then ''F'' contains all ''n''th roots of unity, because these are (by definition) the ''n'' (not necessarily distinct) zeroes of the polynomial ''xn'' − 1. A field extension that is contained in an extension generated by the roots of unity is a ''cyclotomic extension'', and the extension of a field generated by all roots of unity is sometimes called its ''cyclotomic closure''. Thus algebraically closed fields are cyclotomically closed. The converse is not true. Even assuming that every polynomial of the form ''xn'' − ''a'' splits into linear factors is not enough to assure that the field is algebraically closed. If a proposition which can be expressed in the language of first-order logic is true for an algebraically closed field, then it is true for every algebraically closed field with the same characteristic. Furthermore, if such a proposition is valid for an algebraically closed field with characteristic 0, then not only is it valid for all other algebraically closed fields with characteristic 0, but there is some natural number ''N'' such that the proposition is valid for every algebraically closed field with characteristic ''p'' when ''p'' > ''N''. Every field ''F'' has some extension which is algebraically closed. Such an extension is called an algebraically closed extension. Among all such extensions there is one and only one ( up to isomorphism, but not unique isomorphism) which is an algebraic extension of ''F'';See Lang's ''Algebra'', §VII.2 or van der Waerden's ''Algebra I'', §10.1. it is called the algebraic closure of ''F''. The theory of algebraically closed fields has quantifier elimination.Notes
References
* * * * {{DEFAULTSORT:Algebraically Closed Field Field (mathematics)