Irreducibility and Gauss's lemma

For a nonzero integer polynomial, its content is the positive greatest common divisor of its coefficients. Dividing by the content leaves a primitive polynomial whose coefficients have no common prime factor. For example, the polynomial six x squared plus nine x plus three has content three and primitive part two x squared plus three x plus one. The constant factor carries integer divisibility information, while the primitive part carries the positive-degree factorization problem over the rationals.

Gauss's lemma says that the product of primitive integer polynomials is primitive. Suppose instead a prime divided every coefficient of their product. Reducing modulo that prime would give two nonzero polynomials with zero product over a field, impossible because a polynomial ring over a field is a domain. Applying this to primitive parts yields multiplicativity of content. The proof ties integer divisibility to a clean zero-product statement in a simpler coefficient system.

For a primitive integer polynomial, a factorization into positive-degree rational polynomials exists exactly when such a factorization exists with integer coefficients. To see the mechanism, clear denominators from each rational factor, then extract each resulting integer polynomial's content. The primitive-product property controls the scalar introduced during clearing. It prevents an unnoticed common coefficient factor from obstructing the transfer back to an integer factorization.

Irreducibility must still name the coefficient ring. The polynomial x squared minus two is irreducible over the rationals but splits over the reals. A nonzero constant is a unit over the rationals, so a rational scalar is not a nontrivial rational-polynomial factor. For a nonprimitive integer polynomial, first remove its content before comparing irreducibility as an element of the integer polynomial ring with the positive-degree question over the rationals.

Eisenstein's criterion uses a prime p that divides every coefficient except the leading coefficient, while its square does not divide the constant coefficient. If a factorization into two positive-degree integer polynomials existed, reduction modulo p would preserve both degrees because their leading coefficients multiply to the original leading coefficient, which is not divisible by p. Their product reduces to a nonzero multiple of a power of x, so both reduced factors must be positive powers of x up to scalar multiples. Both original constant coefficients would therefore be divisible by p, contradicting the condition on the original constant coefficient. The conclusion is irreducibility over the rationals, with Gauss's lemma supplying the passage from rational factors to integer factors.

Another test reduces a primitive integer polynomial modulo a prime without lowering its degree. If the reduction is irreducible over that finite field, the original polynomial is irreducible over the rationals. The reverse direction is false: an irreducible rational polynomial can become reducible modulo a prime. Also, checking for roots proves irreducibility only in degrees two and three. A degree-four polynomial can have no linear factor yet split into two quadratics, so a root check alone is insufficient there.