IDA: Example

Example

The factorization in irreducibles of X⁴ - 1 in Q[X] is
(X² + 1)(X + 1)(X - 1).
The first factor is irreducible since it has no rational zeros. Considered as a polynomial in C[X] the factorization is
(X + i)(X - i)(X + 1)(X - 1).
Considered as a polynomial in Z/2Z[X], the factorization is

(X + 1)⁴.
As for integers (compare with the example on the factorization record), it is not difficult to verify a factorization. However, it is not always as easy to check whether the found factors are irreducible. A proof that a polynomial f Q[X] with integer coefficients is irreducible, can often be given by computing modulo p for a prime number p. If the polynomial is irreducible modulo p, then it is also irreducible in Z[X].
However, the converse does not hold. There are polynomials f Z[X] which are irreducible but factorize modulo each prime p. An example is f(X) = X⁴ + 1. Modulo 2 it factorizes as (X + 1)⁴ and modulo 3 as (X² - X - 1) (X² + X - 1). It carries too far to show that X⁴ + 1 factorizes modulo every prime.