Algebraically closed field

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In mathematics, a field F is said to be algebraically closed if every polynomial of degree at least 1, with coefficients in F, has a zero (root) in F (i.e. an element x such that the value of the polynomial at x is the additive identity element of F). In that case, every such polynomial splits into linear factors. It can be shown that a field is algebraically closed if and only if it has no proper algebraic extension, and this is sometimes taken as the definition.

As an example, the field of real numbers is not algebraically closed, because the polynomial equation

has no solution in real numbers, even though both of its coefficients (3 and 1) are real. By contrast, the field of complex numbers is algebraically closed: this is stated by the fundamental theorem of algebra.

Every field F has an "algebraic closure", which is the smallest algebraically closed field of which F is a subfield. Each field's algebraic closure is unique up to a non-canonical isomorphism. In particular, the field of complex numbers is an algebraic closure of the field of real numbers. Also, the field of algebraic numbers is the algebraic closure of the field of rational numbers.

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