As an example, the field of real numbers is not algebraically closed, because the polynomial equation
of rational functions with complex coefficients is not closed; for example, the polynomial
Given a field F, the assertion "F is algebraically closed" is equivalent to other assertions: The field F is algebraically closed if and only if the only irreducible polynomials in the polynomial ring F[x] are those of degree one.
The assertion "the polynomials of degree one are irreducible" is trivially true for any field.
If F is algebraically closed and p(x) is an irreducible polynomial of F[x], then it has some root a and therefore p(x) is a multiple of x − a.
On the other hand, if F is not algebraically closed, then there is some non-constant polynomial p(x) in F[x] without roots in F. Let q(x) be some irreducible factor of p(x).
If F has this property, then clearly every non-constant polynomial in F[x] has some root in F; in other words, F is algebraically closed.
On the other hand, that the property stated here holds for F if F is algebraically closed follows from the previous property together with the fact that, for any field K, any polynomial in K[x] can be written as a product of irreducible polynomials.
If every polynomial over F of prime degree has a root in F, then every non-constant polynomial has a root in F.[1] It follows that a field is algebraically closed if and only if every polynomial over F of prime degree has a root in F. The field F is algebraically closed if and only if it has no proper algebraic extension.
On the other hand, if F has some proper algebraic extension K, then the minimal polynomial of an element in K \ F is irreducible and its degree is greater than 1.
The field F is algebraically closed if and only if, for each natural number n, every linear map from Fn into itself has some eigenvector.
But if q(x) = xn + an − 1 xn − 1 + ⋯ + a0, then q(x) is the characteristic polynomial of the n×n companion matrix The field F is algebraically closed if and only if 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. If F is algebraically closed then, since the irreducible polynomials in F[x] are all of degree 1, the property stated above holds by the theorem on partial fraction decomposition.
On the other hand, suppose that the property stated above holds for the field F. Let p(x) be an irreducible element in F[x].
If the field F is algebraically closed, let p(x) and q(x) be two polynomials which are not relatively prime and let r(x) be their greatest common divisor.
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.
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.