In mathematics, a field F is said to be algebraically closed if every polynomial in one variable of degree at least 1, with coefficients in F, has a zero (root) in F. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... In mathematics, a coefficient is a constant multiplicative factor of a certain object. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ... In mathematics, a root (or a zero) of a function f is an element x in the domain of f such that f(x) = 0. ...

Examples

As an example, the field of real numbers is not algebraically closed, because the polynomial equation In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

x² + 1 = 0

has no solution in real numbers, even though all its coefficients (1, 0 and 1) are real. The same argument proves that the field of rational numbers is not algebraically closed either. Also, no finite field F is algebraically closed, because if a₁, a₂, …, a_n are the elements of F, then the polynomial In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...

has no zero in F. By contrast, the field of complex numbers is algebraically closed: this is stated by the fundamental theorem of algebra. Another example of an algebraically closed field is the field of (complex) algebraic numbers. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... In mathematics, the fundamental theorem of algebra states that every complex polynomial in one variable and of degree  â‰¥  has some complex root. ... In mathematics, an algebraic number is any number that is a root of an algebraic equation, a non-zero polynomial with integer (or equivalently, rational) coefficients. ...

Equivalent properties

Given a field F, the assertion “F is algebraically closed” is equivalent to each one of the following:

• Every polynomial p(x) of degree n ≥ 1, with coefficients in F, splits into linear factors. In other words, there are elements k, x₁, x₂, …, x_n of the field F such that

• The field F has no proper algebraic extension.

• For each natural number n, every linear map from Fⁿ into itself has some eigenvector.

• 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.

In mathematics, a coefficient is a constant multiplicative factor of a certain object. ... ... In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In linear algebra, the eigenvectors (from the German eigen meaning own) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In mathematics, a rational function in algebra is a function defined as a ratio of polynomials. ...

Other properties

If F is an algebraically closed field, a is an element of F, and n is a natural number, then a has an n^th root in F, since this is the same thing as saying that the equation xⁿ − a = 0 has some root in F. However, there are fields in which every element has an n^th root (for each natural number n) but which are not algebraically closed. In fact, 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.

Assuming Zorn's lemma, every field F has a unique algebraic closure, which is the smallest algebraically closed field of which F is a subfield. Zorns lemma, also known as the Kuratowski-Zorn lemma, is a proposition of set theory that states: Every non-empty partially ordered set in which every chain (i. ... In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. ...

References

• S. Lang, Algebra, Springer-Verlag, 2004, ISBN 0-387-95385-X

• B. L. van der Waerden, Algebra I, Springer-Verlag, 1991, ISBN 0-387-97424-5