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primitive polynomial

primitive polynomial noun: a polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once; (more specifically) a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.

Noun

primitive polynomial (plural primitive polynomials)

  1. (ring theory) A polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once; (more specifically) a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.
    • We claim that every primitive polynomial can be written as a product of irreducible elements in #92;mathbfD#91;x#93;.[…]By induction on the degree of the primitive polynomials, we conclude that both g(x),h(x) can be written as product of irreducible elements in #92;mathbfD#91;x#93;.1992, T. T. Moh, Algebra, World Scientific, page 124
    • If f(x)#92;in#92;Z#91;x#93;, the ring of polynomials with coefficients in #92;Z, then the content of f, denoted by c(f), is the greatest common divisor of the coefficients of f. The polynomial f(x) is called a primitive polynomial if c(f)#61;1. Since c(fg)#61;c(f)c(g), by Gauss's lemma [Hungerford...2000, David M. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 114
  2. (field theory) A polynomial over a given finite field whose roots are primitive elements; especially, the minimal polynomial of a primitive element of said finite field.
    • Primitive polynomials make the initialization of LFSRs a simpler task since any nonzero state guarantees that all non-zero states will be visited in the maximum length sequence.2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69
    • Definition 7.2 Let f(x) be a monic polynomial of degree n over #92;mathbb#123;F#125;#95;q. If f(x) has a primitive element of #92;mathbb#123;F#125;#95;#123;qⁿ#125; as one of its roots, f(x) is called a primitive polynomial of degree n over #92;mathbb#123;F#125;#95;q.
    • Theorem 7.7 For any positive integer n there always exist primitive polynomials of degree n over #92;mathbb#123;F#125;#95;q. All the n roots of a primitive polynomial of degree n over #92;mathbb#123;F#125;#95;q are primitive elements of #92;mathbb#123;F#125;#95;#123;qⁿ#125;. All primitive...2003, Zhe-Xian Wan, Lectures on Finite Fields and Galois Rings, World Scientific, page 145

primitive polynomial meaning in Tamil

primitive polynomial in Tamil

தொடக்கநிலை அடுக்குக்கோவை

Examples of primitive polynomial in a sentence

  • The polynomial f(x) is called a primitive polynomial if c(f)#61;1.2000, David M. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 114:
  • According to the Gauss lemma, the product of primitive polynomials is primitive.2000, Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, American Mathematical Society, page 1:
  • The HMPC (1992) asserts that, with some (mostly obvious) exceptions, there exists a primitive polynomial of degree n over any finite field with any coefficient arbitrarily prescribed.2008, Stephen D. Cohen, Mateja Preŝern, “The Hansen-Mullen Primitivity Conjecture: Completion of Proof”, in James McKee, Chris Smyth...

Forms of primitive polynomial

pluralprimitive polynomials

Questions about primitive polynomial

What does primitive polynomial mean?
Primitive polynomial means a polynomial over an integral domain R such that no noninvertible element of R divides all its coefficients at once; (more specifically) a polynomial over a GCD domain R such that the greatest common divisor of its coefficients equals 1.
What is primitive polynomial in Tamil?
Primitive polynomial in Tamil is தொடக்கநிலை அடுக்குக்கோவை. See Primitive polynomial meaning in Tamil.
How do you use primitive polynomial in a sentence?
The polynomial f(x) is called a primitive polynomial if c(f)#61;1.
What part of speech is primitive polynomial?
Primitive polynomial is a noun.
What is the plural of primitive polynomial?
The plural of primitive polynomial is primitive polynomials.

Words near primitive polynomial in the dictionary

  1. Primitive Irish
  2. Primitive Methodist
  3. Primitive Methodists
  4. primitive notion
  5. primitive notions
  6. primitive polynomial
  7. primitive polynomials
  8. primitive recursion
  9. primitive recursive
  10. primitive reflex
  11. primitive reflexes

Source and license

Adapted from Wiktionary: primitive polynomial (page history and authors); primitive polynomial (Tamil Wiktionary) (page history and authors). Text available under the Creative Commons Attribution-ShareAlike 4.0 license. Word.ws reformatted the entry and added links. Wiktionary data of 2026-09-02.

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