primitive polynomial

Noun
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 * 1)  A polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once;  a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.
 * 2)  A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.
 * 1)  A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.
 * 1)  A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.
 * 1)  A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.

Usage notes

 * Since fields are rings, the domain of applicability of the ring theory definition includes that of the one specific to Galois fields. It is thus perfectly feasible for a given instance to be a primitive polynomial in both senses of the term: such is the case, for example, for the minimal polynomial (over a given finite field) of a primitive element (i.e., that has said primitive element as root).
 * More precisely, a primitive polynomial over (with coefficients in) $$\mathbb{F}_q$$ of order $$n$$ has roots that are primitive elements of $$\mathbb F_{q^n}$$.
 * Given a primitive element $$\alpha\in\mathbb{F}_q$$, the set of powers $$\{1, \alpha,\dots \alpha^{n-1}\}$$ constitutes a polynomial basis of $$\mathbb{F}_{q^n}$$.
 * In consequence, a primitive polynomial is sometimes defined as a polynomial that generates $$\mathbb F_{q^n}$$.
 * In consequence, a primitive polynomial is sometimes defined as a polynomial that generates $$\mathbb F_{q^n}$$.

Hyponyms

 * polynomial