Gauss's Lemma (polynomial)

Gauss's Lemma (polynomial)

In algebra, in the theory of polynomials (a subfield of ring theory), Gauss's lemma is either of two related statements about polynomials with integer coefficients:

• The first result states that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is called primitive if the greatest common divisor of its coefficients is 1).
• The second result states that if a polynomial with integer coefficients is irreducible over the integers, then it is also irreducible if it is considered as a polynomial over the rationals.

This second statement is a consequence of the first (see proof below). The first statement and proof of the lemma is in Article 42 of Carl Friedrich Gauss's Disquisitiones Arithmeticae (1801).