Definition:Content of Polynomial

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Definition

Integer Polynomial

Let $f \in \Z \left[{X}\right]$ be a polynomial.

Then the content of $f$, $\operatorname{cont} \left({f}\right)$ is the greatest common divisor of the coefficients of $f$.


Rational Polynomial

If $f \in \Q \left[{X}\right]$ then there is some $n \in \N$ such that $n f \in \Z \left[{X}\right]$.

Then we define the content of $f$ to be:

$\operatorname{cont} \left({f}\right) := \dfrac {\operatorname{cont} \left({n f}\right)} n$


Polynomial over a GCD Domain

Let $D$ be a GCD domain.

Let $K$ be the quotient field of $D$.

Let $f \in K \left[{X}\right]$ be a polynomial.

Let $a \in D$ be such that $af \in D \left[{X}\right]$.

Let $d$ be the greatest common divisor of the coefficients of $af$.

Then we define the content of $f$ to be:

$\operatorname{cont} \left({f}\right) := \dfrac d a$


General ring with unity

Let $R$ be a commutative ring with unity.

Let $f \in R[X]$ be a polynomial.


The content of $f$ is the ideal generated by its coefficients.


Also denoted as

$\operatorname{cont} \left({f}\right)$ is also seen denoted $c_f$ or $\left\langle\hspace{-4mu}\left\langle{f}\right\rangle\hspace{-4mu}\right\rangle$.


Also see


Sources