Algebraically closed field

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 in 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 x2 + 1 has no real zero. By contrast, the field of complex numbers is algebraically closed, which is the content of 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 isomorphism. In particular, the field of complex numbers is the algebraic closure of the field of real numbers.


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