Irreducible polynomials

An irreducible polynomial is a polynomial that cannot be factored into polynomials of lower degree with coefficients in a given set (like integers or real numbers), except for multiplying by constants. For example, over the integers, $x^2 + 1$ cannot be factored into polynomials with integer coefficients, so it is irreducible over the integers.

Important properties

• Irreducibility depends on the set of coefficients (for example, a polynomial might be irreducible over the integers but reducible over the real numbers).

• If a polynomial can be written as a product of two polynomials of lower degree, it is reducible.

• All polynomials of degree 1 are irreducible.

• Irreducible polynomials are like 'prime numbers' for polynomials.