4. Polynomials over a field

Polynomial

An expression

    \[a_0+a_1x+a_2x^2+\ldots+a_nx^n,\]

is called a polynomial with variable X over field K,
where a_0,\ldots a_n\in K (and a_n\neq 0). Number n is the degree of the polynomial, and the set of all polynomials with coefficients in K is denoted by K[x]. Two polynomials can be added or multiplied in a natural way.

Every polynomial is related to the polynomial function K\to K associating with a\in K the value of the polynomial for x=a.

Roots of polynomials

An element a\in K is called a root of polynomial f\in K[x], if f(a)=0.

The field K is algebraically closed, if every polynomial over K has a root. The Fundamental Theorem of Algebra states that field \mathbb{C} is algebraically closed.

If f is a polynomial of degree n and c is its root, then f=(x-c)g(x) for a polynomial g(x) of degree n-1.