Definition:Primitive Polynomial (Ring Theory)
This page is about primitive polynomial in the context of ring theory. For other uses, see primitive.
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 $1$.
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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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): primitive polynomial
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): primitive polynomial
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): primitive polynomial