Properties of Content

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Theorem

Let $f, g \in \Q \sqbrk X$ and $q \in \Q$.

The content of a polynomial satisfies the following:


Content of Scalar Multiple

$\cont {q f} = q \cont f$


Content of Monic Polynomial

If $f$ is monic, then $\cont f = \dfrac 1 n$ for some integer $n$.


Polynomial has Integer Coefficients iff Content is Integer

$f$ has integer coefficients if and only if $\cont f$ is an integer.


Content of Rational Polynomial is Multiplicative

Let $h \in \Q \sqbrk X$ be a polynomial with rational coefficients.

Let $\cont h$ denote the content of $h$.


Then for any polynomials $f, g \in \Q \sqbrk X$ with rational coefficients:

$\cont {f g} = \cont f \cont g$


Content of Polynomial in Dedekind Domain is Multiplicative

Let $R$ be a Dedekind domain.

Let $f, g \in R \sqbrk X$ be polynomials.

Let $\cont f$ denote the content of $f$.


Then $\cont {f g} = \cont f \cont g$ is the product of $\cont f$ and $\cont g$.