# Definition:Primitive Polynomial (Ring Theory)

(Redirected from Definition:Primitive Polynomial over Ring)

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## 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

## 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): Entry:**primitive polynomial**