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:

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). This statements have several generalizations described below.

The notion of primitive polynomial used here (which differs from the notion with the same name in the context of finite fields) is defined in any polynomial ring R[X] where R is a commutative ring: a polynomial P in R[X] is primitive if the only elements of R that divide all coefficients of P at once are the invertible elements of R. In the case where R is the ring Z of the integers, this is equivalent to the condition that no prime number divides all the coefficients of P. The notion of irreducible element is defined in any integral domain: an element is irreducible if it is not invertible and cannot be written as a product of two non-invertible elements. If R is an integral domain then so is the polynomial ring R[X] because the leading coefficient of product of non-zero polynomials in R[X] is equal to the product of their leading coefficients, hence is nonzero. A non-constant irreducible polynomial in R[X] is one that is not a product of two non-constant polynomials and which is primitive (because being primitive excludes precisely non-invertible constant polynomials as factors). Note that an irreducible element of R is still irreducible when viewed as constant polynomial in R[X]; this explains the need for "non-constant" above, and in the irreducibility statements below.

The two properties of polynomials with integer coefficients can now be formulated formally as follows:

Primitivity statement: The set of primitive polynomials in Z[X] is closed under multiplication: if P and Q are primitive polynomials then so is their product PQ.

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