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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 $(x + b) (x + c)$ .

Example 2:

The polynomial

$x^{2} - 2$

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

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

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!)