In algebra, Gauss's lemma,[1] named after Carl Friedrich Gauss, is a theorem[note 1] about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic).
Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials.
[note 2]) A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers.
More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers.
In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R".
This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain.
Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be reduced to similar computations on integers and primitive polynomials.
Gauss's lemma, as well as its consequences that do not involve the existence of a complete factorization, remain true over any GCD domain (an integral domain over which greatest common divisors exist).
If one calls primitive a polynomial such that the coefficients generate the unit ideal, Gauss's lemma is true over every commutative ring.
is called primitive if the greatest common divisor of all the coefficients
is 1; in other words, no prime number divides all the coefficients.
Proof: Clearly the product f(x)g(x) of two primitive polynomials has integer coefficients.
Therefore, if it is not primitive, there must be a prime p which is a common divisor of all its coefficients.
Therefore, the coefficients of the product can have no common divisor and are thus primitive.
Note that an irreducible element of Z (a prime number) is still irreducible when viewed as constant polynomial in Z[X]; this explains the need for "non-constant" in the statement.
Gauss's lemma holds more generally over arbitrary unique factorization domains.
There the content c(P) of a polynomial P can be defined as the greatest common divisor of the coefficients of P (like the gcd, the content is actually a set of associate elements).
Irreducibility statement: Let R be a unique factorization domain and F its field of fractions.
be a unique factorization domain with field of fractions
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 (p + qX)(a + qX) = pa + (p+a)qX + q2X2 = q(b + (p+a)X + qX2) shows the failure of the primitivity statement.
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[i√5]).
Another well-known example is the polynomial X2 − X − 1, whose roots are the golden ratio φ = (1 + √5)/2 and its conjugate (1 − √5)/2 showing that it is reducible over the field Q[√5], although it is irreducible over the non-UFD Z[√5] which has Q[√5] as field of fractions.
In the latter example the ring can be made into an UFD by taking its integral closure Z[φ] in Q[√5] (the ring of Dirichlet integers), over which X2 − X − 1 becomes reducible, but in the former example R is already integrally closed.
is a Bézout domain), this agrees with the usual definition of a primitive polynomial.
or it contains a constant polynomial as a factor, the second possibility is ruled out by the assumption.
So, assume otherwise; then there is a non-unit element dividing the coefficients of
It follows from Gauss's lemma that for each unique factorization domain
Gauss's lemma can also be used to show Eisenstein's irreducibility criterion.
Finally, it can be used to show that cyclotomic polynomials (unitary units with integer coefficients) are irreducible.
A similar argument shows: The irreducibility statement also implies that the minimal polynomial over the rational numbers of an algebraic integer has integer coefficients.