Throughout this section we assume that \(A\) is a UFD and \(A[x]\) is a polynomial ring over \(A\) in one variable. We start with the following definition.
Definition7.1.1.(Content of a polynomial over UFD).
The content of a nonzero polynomial\(f(x)=a_0+a_1x+\cdots+a_nx^n\in A[x]\) is \(\gcd(a_0,a_1,\ldots,a_n)\text{.}\)
Convention7.1.2.
The content of \(0\neq f\in A[x]\) is unique up to units. For notational convenience we write \(C(f)\) for a choice of the content of \(f\text{.}\)
Definition7.1.3.(Primitive polynomial).
A polynomial \(f\in A[x]\) is said to be primitive if the content of \(f\text{,}\)\(C(f)=1\text{.}\)
Lemma7.1.4.
Let \(f(x)=a_0+a_1x+\cdots+a_nx^n\in A[x]\text{.}\)
For \(\alpha\in A\text{,}\)\(C(\alpha f)=\alpha C(f)\text{.}\)
There exists a primitive polynomial \(f_1\in A[x]\) such that \(f=C(f)f_1\) and \(\deg f=\deg f_1\text{.}\)
If \(f\) is a nonconstant irreducible polynomial then \(f\) is primitive. The converse is not true.
We only prove the last statement. Assume that \(f\) is a nonconstant irreducible polynomial. We have \(f=C(f)f_1\text{,}\) where \(f_1\) is a primitive polynomial. If \(C(f)\not\sim 1\in A\) then, as \(A\) is a UFD, \(C(f)\) can be factored into a product of irreducibles in \(A\text{.}\) Since irreducible elements of \(A\) remains irreducible in \(A[x]\text{,}\) we must have \(C(f)\sim 1\text{.}\) Thus we get the result.
The converse is not true can be seen by considering the polynomial \(x^2-1\in A[x]\text{.}\)
Lemma7.1.5.(Clearing denominators).
Let \(A\) be a UFD and \(F\) be the field of fractions of \(A\text{.}\) For any \(0\neq f\in F[x]\) there exists \(\gamma\in F\) and a primitive polynomial \(f_1\in A[x]\) such that \(f=\gamma f_1\text{.}\) This factorization is unique up to unit multipliers in \(A\text{.}\)
Suppose that \(f=a_0/b_0+(a_1/b_1)x+\cdots+(a_n/b_n)x^n\text{.}\) Put \(b=b_0b_1\cdots b_n\in A\text{.}\) Then \(bf\in A[x]\text{,}\) say \(bf=f_1\text{.}\) By Lemma 7.1.4 there exists a primitive polynomial \(f_1\) such that \(bf=cf_1\text{.}\) Therefore, \(f=(c/b)f_1\text{.}\)
Now assume that there exists primitive polynomials \(f_1,f_2\in A[x]\) and \(\gamma_1=c/b,\gamma_2=d/e\in F\) such that \(f=\gamma_if_i\) for \(i=1,2\text{.}\) Hence \(ce f_1=bd f_2\in A[x]\text{.}\) As \(f_i\) are primitive we must have \(ce\sim bd\text{,}\) i.e., there exists a unit \(u\in A\) such that \(ce=ubd\text{,}\) i.e., \(c/b=u\cdot d/e\text{.}\) Hence uniqueness is proved.
Corollary7.1.6.
If \(f,g\in A[x]\) are primitive polynomials and \(f\sim g\) in \(F[x]\text{.}\) Then \(f\sim g\) in \(A[x]\text{.}\)
Suppose that \(0\neq \gamma\in F\) is such that \(f=\gamma g\text{.}\) By uniqueness part of Lemma 7.1.5, there exists a unit \(u\in A\) such that \(1=u\cdot\gamma\text{.}\) Thus \(\gamma\in A\) is a unit.
Corollary7.1.7.
Let \(f\in A[x]\) be a nonconstant primitive. If \(f\) is reducible polynomial in \(A[x]\text{,}\) then there exists \(g,h\in A[x]\) with \(0\lt\deg g\lt\deg f\) and \(0\lt\deg h\lt\deg f\text{.}\)
Suppose that there are \(g,h\in A[x]\) such that \(f=gh\text{.}\) If \(\deg g=0\text{,}\) i.e., if \(g\) is constant then \(1\sim C(f)\sim gC(h)\text{,}\) i.e., \(g\) is a unit and \(f\sim h\) in \(A[x]\text{.}\)
Theorem7.1.8.(Gauss’s theorem).
Let \(A\) be a UFD. The product of primitive polynomials in \(A[x]\) is primitive.
Let \(f,g\in A[x]\) be primitive polynomials. Suppose that \(fg\) is not a primitive polynomial. Let \(p\in A\) be a prime element in \(A\) which is a factor of \(C(fg)\text{.}\) Thus \((R/p)[x]\) is an integral domain and by the assumption \(0=\overline{f}\overline{g}\in (R/p)[x]\text{.}\) Hence either \(\overline{f}=0\) or \(\overline{g}=0\text{.}\) In particular, either \(p\mid C(f)\) or \(p\mid C(g)\text{.}\) This is a contradiction to the assumption that \(f\) and \(g\) are primitive polynomials.
