Polynomial content

The content of a polynomial is a greatest common divisor of its coefficients, up to a unit.

For a nonzero polynomial $g\in \mathbb{Q}[t]$, choose a positive integer $n$ clearing its denominators and set $c(g)=c(ng)/n$, where the polynomial content of an integer-coefficient polynomial is the positive greatest common divisor of its coefficients. This is independent of $n$: if $f/n=h/m$, then $mf=nh$, hence $mc(f)=nc(h)$.

The integer-coefficient product formula from Gauss lemma for polynomials givesMoreover $g/c(g)$ is an integer-coefficient primitive polynomial: if $g=f/n$, then $g/c(g)=f/c(f)$. For example,This normalization proves a useful divisibility descent. If $f\in \mathbb{Z}[t]$ is primitive, $g\in \mathbb{Z}[t]$ and $g=fq$ for $q\in \mathbb{Q}[t]$, then $c(q)=c(g)$ is an integer. Since $q/c(q)\in \mathbb{Z}[t]$, it follows that $q\in \mathbb{Z}[t]$.

A primitive polynomial has unit content.

Over a UFD, products of primitive polynomials are primitive and factorization over the fraction field descends after content is removed.

For a field with a Non-Archimedean absolute value, define the coefficient norm $\Vert f{\Vert }_G={\max }_i|a_i|$ for $f(T)={\sum }_i{a}_i{T}^i$. Then $\Vert fg{\Vert }_G=\Vert f{\Vert }_G\Vert g{\Vert }_G$. For nonzero polynomials, scale each by one of its coefficients of maximum absolute value; the resulting polynomials have integral coefficients and at least one unit coefficient. Their reductions in the residue field are nonzero, so their product is nonzero. The product therefore also has coefficient norm $1$, proving the formula after reversing the scaling. This compares binary-polynomial coefficient heights with the heights of their roots.

If $R$ is a unique factorization domain with fraction field $F$, then $R[X]$ is a unique factorization domain. A primitive polynomial irreducible in $R[X]$ remains irreducible, hence prime, in the principal ideal domain $F[X]$; Gauss lemma for polynomials brings divisibility back to $R[X]$.

The polynomial ring $\mathbb{Z}[X]$ is a unique factorization domain by Gauss lemma for polynomials, but its ideal $(2,X)$ is not principal. Any generator would divide both 2 and $X$, hence be a unit: a divisor of 2 has degree zero, and a constant dividing $X$ is $\pm 1$. Yet evaluation at zero followed by reduction modulo 2 annihilates the ideal and not 1. It embeds in the principal ideal domain $\mathbb{Q}[X]$, illustrating that the property need not pass to a subring.

Every rational function in $\mathbb{C}(X,Y)$ fixed by exchanging $X$ and $Y$ is a quotient of coprime symmetric polynomials. In a coprime presentation $p/q$, symmetry givesUnique factorization implies $p^{\ast }=\lambda p$ and $q^{\ast }=\lambda q$. The alternative $\lambda =-1$ would make both divisible by $X-Y$, contradicting coprimality, so $\lambda =1$.