is called a polynomial with variable over field ,
where (and ). Number is the degree of the polynomial, and the set of all polynomials with coefficients in is denoted by . Two polynomials can be added or multiplied in a natural way.
Every polynomial is related to the polynomial function associating with the value of the polynomial for .
Roots of polynomials
An element is called a root of polynomial , if .
The field is algebraically closed, if every polynomial over has a root. The Fundamental Theorem of Algebra states that field is algebraically closed.
If is a polynomial of degree and is its root, then for a polynomial of degree .