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"ref": "1992, T. T. Moh, Algebra, World Scientific, page 124",
"text": "We claim that every primitive polynomial can be written as a product of irreducible elements in #x5C;mathbfD#x5B;x#x5D;.[…]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 #x5C;mathbfD#x5B;x#x5D;.",
"type": "quotation"
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"ref": "2000, David M. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 114",
"text": "If f(x)#x5C;in#x5C;Z#x5B;x#x5D;, the ring of polynomials with coefficients in #x5C;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)#x3D;1. Since c(fg)#x3D;c(f)c(g), by Gauss's lemma [Hungerford, 74], the set S of primitive polynomials in #x5C;Z#x5B;x#x5D; is a multiplicatively closed set. Define #x5C;Lambda#x3D;#x5C;Z#x5B;x#x5D;#x5F;S, the localization of #x5C;Z#x5B;x#x5D; at S, a subring of the field of quotients #x5C;Q(x) of #x5C;Z#x5B;x#x5D;. Elements of #x5C;Lambda are of the form f(x)#x2F;g(x) with f(x),g(x)#x5C;in#x5C;Z#x5B;x#x5D; and g(x) a primitive polynomial.",
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"ref": "2000, Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, American Mathematical Society, page 1",
"text": "According to the Gauss lemma, the product of primitive polynomials is primitive. Therefore if f(x),g(x) are primitive and f(x)#x3D;g(x)h(x) with h(x) in F#x5B;x#x5D;, then necessarily h(x) is in A#x5B;x#x5D; and primitive. […]The irreducible elements of A#x5B;x#x5D; are irreducible elements of A and primitive polynomials which are irreducible in F#x5B;x#x5D;.",
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"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."
],
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"_dis1": "72 28",
"sense": "polynomial over an integral domain such that no noninvertible element divides all of its coefficients",
"word": "monic polynomial"
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"(algebra, 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."
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"ref": "2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69",
"text": "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.",
"type": "quotation"
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"ref": "2003, Zhe-Xian Wan, Lectures on Finite Fields and Galois Rings, World Scientific, page 145",
"text": "Definition 7.2 Let f(x) be a monic polynomial of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q. If f(x) has a primitive element of #x5C;mathbb#x7B;F#x7D;#x5F;#x7B;qⁿ#x7D; as one of its roots, f(x) is called a primitive polynomial of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q.\nTheorem 7.7 For any positive integer n there always exist primitive polynomials of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q. All the n roots of a primitive polynomial of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q are primitive elements of #x5C;mathbb#x7B;F#x7D;#x5F;#x7B;qⁿ#x7D;. All primitive polynomials of degree n over #x5C;mathbbF#x5F;q are irreducible over #x5C;mathbb#x7B;F#x7D;#x5F;q. The number of primitive polynomials of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q is equal to #x5C;phi(qⁿ-1)#x2F;n.",
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"ref": "2008, Stephen D. Cohen, Mateja Preŝern, “The Hansen-Mullen Primitivity Conjecture: Completion of Proof”, in James McKee, Chris Smyth, editors, Number Theory and Polynomials, Cambridge University Press, page 89",
"text": "This paper completes an efficient proof of the Hansen-Mullen Primitivity Conjecture (HMPC) when n = 5, 6, 7 or 8. 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.",
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"glosses": [
"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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"(algebra, 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."
],
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"_dis1": "34 66",
"english": "of a polynomial",
"word": "primitive part"
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"_dis1": "34 66",
"english": "of a polynomial",
"word": "content"
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"word": "irreducible"
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"english": "of a polynomial",
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"ref": "1992, T. T. Moh, Algebra, World Scientific, page 124",
"text": "We claim that every primitive polynomial can be written as a product of irreducible elements in #x5C;mathbfD#x5B;x#x5D;.[…]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 #x5C;mathbfD#x5B;x#x5D;.",
"type": "quotation"
},
{
"ref": "2000, David M. Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 114",
"text": "If f(x)#x5C;in#x5C;Z#x5B;x#x5D;, the ring of polynomials with coefficients in #x5C;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)#x3D;1. Since c(fg)#x3D;c(f)c(g), by Gauss's lemma [Hungerford, 74], the set S of primitive polynomials in #x5C;Z#x5B;x#x5D; is a multiplicatively closed set. Define #x5C;Lambda#x3D;#x5C;Z#x5B;x#x5D;#x5F;S, the localization of #x5C;Z#x5B;x#x5D; at S, a subring of the field of quotients #x5C;Q(x) of #x5C;Z#x5B;x#x5D;. Elements of #x5C;Lambda are of the form f(x)#x2F;g(x) with f(x),g(x)#x5C;in#x5C;Z#x5B;x#x5D; and g(x) a primitive polynomial.",
"type": "quotation"
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"ref": "2000, Jun-ichi Igusa, An Introduction to the Theory of Local Zeta Functions, American Mathematical Society, page 1",
"text": "According to the Gauss lemma, the product of primitive polynomials is primitive. Therefore if f(x),g(x) are primitive and f(x)#x3D;g(x)h(x) with h(x) in F#x5B;x#x5D;, then necessarily h(x) is in A#x5B;x#x5D; and primitive. […]The irreducible elements of A#x5B;x#x5D; are irreducible elements of A and primitive polynomials which are irreducible in F#x5B;x#x5D;.",
"type": "quotation"
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"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."
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"(algebra, 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."
],
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"ref": "2002, Charles E. Stroud, A Designer's Guide to Built-in Self-Test, Kluwer Academic, page 69",
"text": "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.",
"type": "quotation"
},
{
"ref": "2003, Zhe-Xian Wan, Lectures on Finite Fields and Galois Rings, World Scientific, page 145",
"text": "Definition 7.2 Let f(x) be a monic polynomial of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q. If f(x) has a primitive element of #x5C;mathbb#x7B;F#x7D;#x5F;#x7B;qⁿ#x7D; as one of its roots, f(x) is called a primitive polynomial of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q.\nTheorem 7.7 For any positive integer n there always exist primitive polynomials of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q. All the n roots of a primitive polynomial of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q are primitive elements of #x5C;mathbb#x7B;F#x7D;#x5F;#x7B;qⁿ#x7D;. All primitive polynomials of degree n over #x5C;mathbbF#x5F;q are irreducible over #x5C;mathbb#x7B;F#x7D;#x5F;q. The number of primitive polynomials of degree n over #x5C;mathbb#x7B;F#x7D;#x5F;q is equal to #x5C;phi(qⁿ-1)#x2F;n.",
"type": "quotation"
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{
"ref": "2008, Stephen D. Cohen, Mateja Preŝern, “The Hansen-Mullen Primitivity Conjecture: Completion of Proof”, in James McKee, Chris Smyth, editors, Number Theory and Polynomials, Cambridge University Press, page 89",
"text": "This paper completes an efficient proof of the Hansen-Mullen Primitivity Conjecture (HMPC) when n = 5, 6, 7 or 8. 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.",
"type": "quotation"
}
],
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"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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"(algebra, 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."
],
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"wikipedia": [
"Primitive polynomial"
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