Gauss's Lemma (polynomial) - Formal Statements

Formal Statements

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 where R is an integral domain: a polynomial P in R 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. In the case of a polynomial ring R, this means that a non-constant irreducible polynomial 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; 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 is closed under multiplication: if P and Q are primitive polynomials then so is their product PQ.

• Irreducibility statement: A non-constant polynomial in Z is irreducible in Z if and only if it is both irreducible in Q and primitive in Z.

These statements can be generalized to any unique factorization domain (UFD), where they become

• Primitivity statement: If R is a UFD, then the set of primitive polynomials in R is closed under multiplication.

• Irreducibility statement: Let R be a UFD and F its field of fractions. A non-constant polynomial in R is irreducible in R if and only if it is both irreducible in F and primitive in R.

The condition that R is a UFD is not superfluous. In a ring where factorization is not unique, say pa = qb with p and q irreducible elements that do not divide any of the factors on the other side, the product

shows the failure of the primitivity statement. For a concrete example one can take

In this example the polynomial 3 + 2X + 2X2 (obtained by dividing the right hand side by q = 2) provides an example of the failure of the irreducibility statement (it is irreducible over R, but reducible over its field of fractions ℚ). Another well known example is the polynomial X2 − X − 1, whose roots are the golden ratio and its conjugate, showing that it is reducible over the field ℚ, although it is irreducible over the non-UFD ℤ which has ℚ as field of fractions. In the latter example the ring can be made into an UFD by taking its integral closure ℤ in ℚ (the ring of Dirichlet integers), over which X2 − X − 1 becomes reducible, but in the former example R is already integrally closed.