Algebraically closed field

In mathematics, a field $F$ is algebraically closed if every non-constant polynomial in $F [x]$ (the univariate polynomial ring with coefficients in $F$ ) has a root in $F$ .

Examples[edit]

As an example, the field of real numbers is not algebraically closed, because the polynomial equation $x^{2}+1=0$ has no solution in real numbers, even though all its coefficients (1 and 0) are real. The same argument proves that no subfield of the real field is algebraically closed; in particular, the field of rational numbers is not algebraically closed. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed. Another example of an algebraically closed field is the field of (complex) algebraic numbers.

No finite field F is algebraically closed, because if a₁, a₂, ..., a_n are the elements of F, then the polynomial (x − a₁)(x − a₂) ⋯ (x − a_n) + 1 has no zero in F. However, the union of all finite fields of a fixed characteristic p is an algebraically closed field, which is, in fact, the algebraic closure of the field $\mathbb {F} _{p}$ with p elements.

Other properties[edit]

If F is an algebraically closed field and n is a natural number, then F contains all nth 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.^[2]

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;^[3] it is called the algebraic closure of F.

The theory of algebraically closed fields has quantifier elimination.