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 (where the field extension is , being the field of complex numbers and being the field of rational numbers).
Properties.
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 (except their own elements) are called algebraically closed. The field of complex numbers is an example.