Primitive polynomial

English

Noun

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1. Template:Lb A polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once; Template:Lb a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.
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2. Template:Lb A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.
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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).
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  • More precisely, a primitive polynomial over (with coefficients in) ${𝔽}_q$ of order $n$ has roots that are primitive elements of ${𝔽}_{q^n}$.
  • Given a primitive element $\alpha \in {𝔽}_q$, the set of powers $\{1,\alpha ,\dots {\alpha }^{n-1}\}$ constitutes a polynomial basis of ${𝔽}_{q^n}$.
    • In consequence, a primitive polynomial is sometimes defined as a polynomial that generates ${𝔽}_{q^n}$.

Hyponyms

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Related terms

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See also

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Further reading

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• Primitive polynomial on Template:W
• Galois field structure on Template:W
• Primitive Polynomial on Template:W