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Algebraically closed field

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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 this 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 which is not algebraically closed can be formally extended by adjoining roots of polynomials without zeros. If one adjoins to F all roots of all polynomials, the resulting field is called the algebraic closure of F. For example, C (the field of complex numbers) is the algebraic closure of R (the field of real numbers).

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