# Irreducible (Prime) Polynomials

A polynomial with integer coefficients that cannot be factored into polynomials of lower degree , also with integer coefficients, is called an irreducible or prime polynomial .

Example 1:

${x}^{2}+x+1$

is an irreducible polynomial. There is no way to find two integers $b$ and $c$ such that their product is $1$ and their sum is also $1$ , so we cannot factor into linear terms $\left(x+b\right)\left(x+c\right)$ .

Example 2:

The polynomial

${x}^{2}-2$

is irreducible over the integers. However, we could factor it as

$\left(x-\sqrt{2}\right)\left(x+\sqrt{2}\right)$

if we are allowed to use irrational numbers. So the irreducibility of a polynomial depends on the number system you're working in.

(When you study complex numbers , you'll find that the only irreducible polynomials over C are the degree $1$ polynomials!)