Properties of Content
Jump to navigation
Jump to search
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 $\cont g$.