Definition:Primitive Polynomial (Ring Theory)

Definition

Let $\Q \sqbrk X$ be the ring of polynomial forms over the field of rational numbers in the indeterminate $X$.

Let $f \in \Q \sqbrk X$ be such that:

$\cont f = 1$

where $\cont f$ is the content of $f$.

That is:

The greatest common divisor of the coefficients of $f$ is equal to $1$.

Then $f$ is described as primitive.

Also defined as

From Polynomial has Integer Coefficients iff Content is Integer it follows that, if $f$ is a primitive polynomial, then:

$f \in \Z \sqbrk X$

where $\Z \sqbrk X$ is the ring of polynomial forms over the integral domain of integers in the indeterminate $X$.

Hence this definition can often be found stated with the additional condition that the coefficients of $f$ are integers, but it should be noted that this condition is in fact superfluous.

Also see

• Product of Primitive Polynomials is Primitive
• Definition:Primitive Part of Polynomial

Sources

• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 31$. Polynomials with Integer Coefficients
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): primitive polynomial