Vipul: New page: ==Definition== Let $R$ be a commutative unital ring and $f \in R[x]$ be a polynomial. The content of $f$ is defined as the ideal of $R$ ...

2009-02-01T16:43:56Z

New page: ==Definition== Let <math>R</math> be a commutative unital ring and <math>f \in R[x]</math> be a polynomial. The content of <math>f</math> is defined as the ideal of <math>R</math>...

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==Definition==

Let <math>R</math> be a [[commutative unital ring]] and <math>f \in R[x]</math> be a polynomial. The content of <math>f</math> is defined as the [[ideal]] of <math>R</math> generated by the coefficients of <math>f</math>.

In the case where <math>R</math> is a [[gcd domain]] (for instance, where <math>R</math> is a [[Bezout domain]] or a [[unique factorization domain]]), the content of <math>f</math> is defined as the greatest common divisor of all the coefficients of <math>f</math>. Note that the greatest common divisor is completely determined (upto associates) by the ideal generated by the coefficients: it is the generator of the smallest principal ideal containing that ideal.

In the case that <math>R</math> is a [[Bezout domain]], the content of <math>f</math> is in fact the generator of the ideal generated by all the coefficients.

==Related notions==

A [[primitive polynomial]] over a ring is a polynomial such that the ideal generated by the coefficients is not contained in any proper principal ideal. Equivalently, the greatest common divisor of the coefficients is <math>1</math>.

Content of a polynomial - Revision history

Vipul: New page: ==Definition== Let R be a commutative unital ring and f \in R[x] be a polynomial. The content of f is defined as the ideal of R...

Vipul: New page: ==Definition== Let $R$ be a commutative unital ring and $f \in R[x]$ be a polynomial. The content of $f$ is defined as the ideal of $R$ ...