There are two related results that are commonly called “Gauss Lemma”. The first is that the product of primitive polynomial is still primitive. The second result is that a primitive polynomial is irreducible over a UFD (Unique Factorization Domain) D, if and only if it is irreducible over its quotient field.
Gauss Lemma: Product of primitive polynomials is primitive
If is a unique factorization domain and , then . In particular, the product of primitive polynomials is primitive.
(Hungerford pg 163)
Write and with , primitive. Consequently
Hence it suffices to prove that is primitive, that is, is a unit. If and , then with .
If is not primitive, then there exists an irreducible element in such that for all . Since is a unit , hence there is a least integer such that
Similarly there is a least integer such that
Since divides must divide . Since every irreducible element in (UFD) is prime, or . This is a contradiction. Therefore is primitive.
Primitive polynomials are associates in iff they are associates in
Let be a unique factorization domain with quotient field and let and be primitive polynomials in . Then and are associates in if and only if they are associates in .
() If and are associates in the integral domain , then for some unit . Since the units in are nonzero constants, so , hence with and . Thus .
Since and are units in ,
Therefore, for some unit and . Consequently, (since ), hence and are associates in .
() Clear, since if for some , then and are associates in .
Primitive is irreducible in iff is irreducible in
Let be a UFD with quotient field and a primitive polynomial of positive degree in . Then is irreducible in if and only if is irreducible in .
() Suppose is irreducible in and with and , . Then and with and , .
Let and for each let If (clear denominators of by multiplying by product of denominators), then with , and primitive.
Verify that and . Similarly with , , primitive and . Consequently, , hence . Since is primitive by hypothesis and is primitive by Gauss Lemma,
This means and are associates in . Thus for some unit . So , hence and are associates in . Consequently is reducible in (since ), which is a contradiction. Therefore, is irreducible in .
() Conversely if is irreducible in and with , then one of , (say ) is a unit in and thus a (nonzero) constant. Thus . Since is primitive, must be a unit in and hence in . Thus is irreducible in .