Algebraic element


In mathematics, if is a field extension of, then an element of is called an algebraic element over, or just algebraic over, if there exists some non-zero polynomial with coefficients in such that. Elements of which are not algebraic over are called transcendental over.
These notions generalize the algebraic numbers and the transcendental numbers.

Examples

The following conditions are equivalent for an element of :
This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over are again algebraic over. The set of all elements of which are algebraic over is a field that sits in between and.
If is algebraic over, then there are many nonzero polynomials with coefficients in such that. However, there is a single one with smallest degree and with leading coefficient 1. This is the minimal polynomial of a and it encodes many important properties of.
Fields that do not allow any algebraic elements over them are called algebraically closed. The field of complex numbers is an example.