Definition:Content of Polynomial
Definition
Integer Polynomial
Let $f \in \Z \sqbrk X$ be a polynomial with integer coefficients.
Then the content of $f$, denoted $\cont f$, is the greatest common divisor of the coefficients of $f$.
Rational Polynomial
Let $f \in \Q \sqbrk X$ be a polynomial with rational coefficients.
The content of $f$ is defined as:
- $\cont f := \dfrac {\cont {n f} } n$
where $n \in \N$ is such that $n f \in \Z \sqbrk X$.
Polynomial in GCD Domain
Let $D$ be a GCD domain.
Let $K$ be the field of quotients of $D$.
Let $f \in K \sqbrk X$ be a polynomial.
Let $a \in D$ be such that $a f \in D \sqbrk X$.
Let $d$ be the greatest common divisor of the coefficients of $a f$.
Then we define the content of $f$ to be:
- $\cont f := \dfrac d a$
Commutative Ring with Unity
Let $R$ be a commutative ring with unity.
Let $f \in R \sqbrk X$ be a polynomial.
The content of $f$ is the ideal generated by its coefficients.
Also denoted as
The content of a polynomial $f$ can be seen in the literature variously denoted as:
- $\cont f$ (currently used on $\mathsf{Pr} \infty \mathsf{fWiki}$)
- $c_f$
- $\left\langle \! \left\langle {f} \right\rangle \! \right\rangle$
Also see
- Definition:Primitive Polynomial (Ring Theory): A polynomial $f$ is primitive if and only if $\cont f = 1$.
- Results about Content of Polynomial can be found here.