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Topic: Irreducible polynomial

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	Math Forum - Ask Dr. Math
		In answer to your question, that does not mean that the polynomial is irreducible.
		That's because the 3rd degree ones divide x^7 + 1, and 7 is not a divisor of 51; and the 4th degree ones divide either x^15 + 1 or x^5 + 1, and neither of these is a divisor of 51.
		Irreducible polynomials of period 17 or 51 have degree 8, because 17 and 51 are factors of 2^8 - 1 = 255, and no lower power of 2 will do.
	mathforum.org /library/drmath/view/52008.html (500 words)

	PlanetMath: proof that the cyclotomic polynomial is irreducible
		, since it splits this polynomial and is generated as an algebra by a single root of the polynomial.
		"proof that the cyclotomic polynomial is irreducible" is owned by djao.
		This is version 6 of proof that the cyclotomic polynomial is irreducible, born on 2002-05-08, modified 2005-04-03.
	planetmath.org /encyclopedia/ProofThatTheCyclotomicPolynomialIsIrreducible.html (258 words)

	A Test for Absolute Irreducibility of Polynomials with Rational Coefficients
		While univariate polynomials with coefficients in a field k can always be factored as products of linear polynomials over the algebraic closure k of k, in the multivariate case irreducible polynomials over k may have arbitrary degree.
		A multivariate polynomial with coefficients in k which is irreducible over k is called absolutely irreducible and the decomposition of a multivariate polynomial as a product of absolutely irreducible polynomials is called its absolute factorization.
		Polynomials whose degree decreases when reduced mod p have to be taken into account, but their quantity does not change the final result much.
	algo.inria.fr /seminars/sem97-98/ragot.html (1193 words)

	Irreducible polynomial - Wikipedia, the free encyclopedia
		Galois theory studies the relationship between a field, its Galois group, and its irreducible polynomials in depth.
		of integers, the first two polynomials are reducible, but the last two are irreducible (the third does not have integer coefficients).
		Hence, all irreducible polynomials are of degree 1.
	en.wikipedia.org /wiki/Irreducible_polynomial (773 words)

	Polynomial Summary
		Because of their simple structure, polynomials are easy to evaluate, and are used extensively in numerical analysis for polynomial interpolation or to numerically integrate more complex functions.
		The degree of a term in a polynomial is the sum of all of the exponents on the variables in that term, where a variable with no exponent is understood to have an exponent of 1.
		In knot theory the Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial are important knot invariants.
	www.bookrags.com /Polynomial (3406 words)

	Polynomials - PineWiki
		Evaluation turns a polynomial into a function from R to R, but we consider polynomials with different coefficients to be different even if they yield the same function (this usually only happens when R is finite, e.g.
		Given a polynomial p(x), the largest exponent of any term with a nonzero coefficient is called the degree of a polynomial; this is often abrreviated as deg(p(x)).
		For polynomials, the role of primes in integer factorization is taken by irreducible polynomials, where a polynomial p is irreducible if p(x) = a(x)b(x) holds only if at lest one of a(x) or b(x) has degree zero.
	pine.cs.yale.edu /pinewiki/Polynomials (1943 words)

	Irreducibility Criteria
		It often happens that this criterion is not directly applicable to a given polynomial f(x), but it may be applicable to f(x+a) for some constant a.
		A primal polynomial F(x) is one for which the greatest common divisor of all the values F(k) is 1.
		Specifically, in 1857 Bouniakowsky conjectured that if f(x) is an irreducible polynomial with integer coefficients such that no number greater than 1 divides all the values of f(k) for every integer k, then f(k) is prime for infinitely many integers k.
	www.mathpages.com /home/kmath406.htm (917 words)

	[No title]
		A primitive polynomial with a minimum number of nonzero coefficients and polynomials be- longing to all possible exponents is given for each degree 17 through 34.
		The reciprocal polynomial of an irreducible polynomial is also irreducible, and the reciprocal polynomial of a primitive poly- nomial is primitive.
		For each degree, a primitive polynomial with a minimum number of nonzero coefficients was chosen, and this polynomial is the first in the table of polynomials of this degree.
	www.cs.umbc.edu /~lomonaco/f97/442/Peterson_Table.html (613 words)

	Finite Fields Package - Wolfram Mathematica
		The internal form of a finite field element is GF[p, ilist][elist] where GF stands for Galois field, p is the prime characteristic of the field, ilist is the coefficient list of the irreducible polynomial which defines multiplication in the field, and elist is the coefficient list of the polynomial representing the particular element.
		When this form is used, the package automatically selects an irreducible polynomial, with coefficients given by ilist and with the property that GF[p, ilist][{0, 1}] is a primitive element of the field.
		Since it is possible to have fields of the same size using different irreducible polynomials, it is useful to be able distinguish elements from these fields.
	reference.wolfram.com /mathematica/FiniteFields/tutorial/FiniteFields.html (2110 words)

	Primitive Polynomial Computation Theory and Algorithm
		The field is exactly the set of all polynomials of degree 0 to n-1 with the two field operations being addition and multiplication of polynomials modulo g(x) and with modulo p integer arithmetic on the polynomial coefficients.
		A primitive polynomial f(x) is irreducible in GF(p).
		A primitive polynomial is the minimal polynomial of a generator, and its roots are conjugates of the generator.
	www.seanerikoconnor.freeservers.com /Mathematics/AbstractAlgebra/PrimitivePolynomials/theory.html (3052 words)

	Primitive polynomial - Wikipedia, the free encyclopedia
		A polynomial over a unique factorization domain (such as the integers) whose greatest common divisor of its coefficients is one.
		Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible.
		Primitive polynomials are used in the representation of elements of a finite field.
	en.wikipedia.org /wiki/Primitive_polynomial (483 words)

	Decoding method and apparatus for cyclic codes - Patent 4677623
		The irreducible polynomial is defined as one which is not divisible without a residue by any polynomial having an order of no less than one and no more than m.sub.i -1.
		The irreducible polynomial P.sub.i (x) and the polynomial (x.sup.c +1) have respective periods, and the syndromes for those polynomials set in the dividers when the input code is received return to their original contents after the numbers of times of shifting corresponding to the periods of the respective polynomials, if the input code is all-zero.
		In the generator polynomial shown by formula (14), the polynomial (x.sup.11 +x.sup.2 +1) is a primitive polynomial and the period e of the FSRP is 2047 (=2.sup.11 -1).
	www.freepatentsonline.com /4677623.html (8080 words)

	DESCRIPTION OF CMAT
		Factorization of a polynomial in Z[x] is accomplished using an algorithm outlined in [Mus].
		Factorization of a polynomial in Q(i)[x] is accomplished using an algorithm outlined in [Tra].
		Irreducible polynomials of given degree (mod p) are constructed using a probabilistic algorithm from [Lu2][145-149].
	www.numbertheory.org /cmat/krm_cmat.html (1632 words)

	Math Forum - Ask Dr. Math
		Date: 10/18/2004 at 02:02:59 From: Gunnar Subject: Irreducible polynomials over Q Let p be a prime number.
		Date: 10/18/2004 at 14:01:32 From: Doctor Pete Subject: Re: Irreducible polynomials over Q Hi Gunnar, This question seems to be solved best by Eisenstein's criterion, which states that the polynomial f(x) = a[n] x^n + a[n-1] x^(n-1) +...
		What makes this question interesting, then, is the idea that a polynomial may not satisfy the Eisenstein criterion as it is written, but that a translation may allow one to show irreducibility.
	mathforum.org /library/drmath/view/67125.html (510 words)

	Springer Online Reference Works
		A polynomial is called absolutely irreducible if it is irreducible over the algebraic closure of its field of coefficients.
		The absolutely irreducible polynomials of a single variable are the polynomials of degree 1.
		Over the field of real numbers any irreducible polynomial in a single variable is of degree 1 or 2 and a polynomial of degree 2 is irreducible if and only if its discriminant is negative.
	eom.springer.de /i/i052620.htm (231 words)

	Factorization
		Given a square-free univariate polynomial p in F[x] with F a finite field, this function returns the distinct-degree factorization of p as a sequence of tuples of length 2, each consisting of a degree d, together with the product of the degree-d irreducible factors of p.
		Given a polynomial p in K[x] such that p is irreducible and K is a field, this function returns true iff p is separable.
		The resultant of univariate polynomials p and q (of degree m and n) in R[x], which is by definition the determinant of the Sylvester matrix for p and q (a matrix of rank m + n containing coefficients of f and g as entries).
	www.math.wisc.edu /help/magma/text380.html (787 words)

	Science and Reason: Failure of unique factorization
		The same kind of argument shows that 1±2√-5 must be irreducible, since both conjugates have norm 21, and any non-unit α that divided either would have a norm equal to 3 or 7, which we just observed is impossible.
		The set of irreducible elements of A is the same as the set of prime elements of A (up to unit factors).
		However, if there are irreducible elements that aren't prime, then factorizations of some integers into powers of irreducibles will not be unique, and some integers will not even have a factorization into powers of primes.
	scienceandreason.blogspot.com /2007/11/failure-of-unique-factorization.html (2538 words)

	Finite fields in MAPLE
		lambda0(n,p) (where the number of irreducible polynomials mod p of degree n with (maximum) period p^n-1 is lambda0(n,p)).
		gf_eval_poly2(f,x,y,a,b,FF) evaluates a polynomial in 2 variables f at x=a,y=b in GF field G; f must have integer coefficients (a,b in GF notation, i.e., as a polynomial in 10^4 with coefficients in 0..p-1).
		This can be explored graphically by plotting the probability that a randomly choosen irreducible polynomial of a fixed degree n has maximum period, n=3,4,5,....,100.
	cadigweb.ew.usna.edu /~wdj/maplestuff/finite_fields.html (1276 words)

	Field extensions
		Irreducible Polynomials - Show that there are exactly (p^2-p)/2 monic irreducible polynomials of degree 2 over Z_p, where p is any prime.
		Using the definition of irreducibility, Theorem: A polynomial of degree 2 or 3 is irr...
		Irreducible palindromic polynomial, irreducible polynomials, number of polynomials.
	www.brainmass.com /homeworkhelp/math/other/57545 (211 words)

	Finite Fields
		The polynomials that have the same remainder after division form equivalence classes, which are the elements of the quotient field.
		The zeros of an irreducible polynomial f(x) in K[x] in the splitting field for f(x) over K are called conjugates.
		Now the primitive elements are to be found among the roots of the irreducible polynomials (they cannot be elements of the prime field).
	www-math.cudenver.edu /~wcherowi/courses/finflds.html (3085 words)

	Irreducible (Prime) Polynomials
		A polynomial with integer coefficients that cannot be factored into polynomials of lower degree, also with integer coefficients, is called an irreducible or prime polynomial.
		There is no way to find two integers b and c such that their product is 1 and their sum is also 1, so we cannot factor into linear terms (x + b)(x + c).
		So the irreducibility of a polynomial depends on the number system you're working in.
	hotmath.com /hotmath_help/topics/irreducible-polynomials.html (101 words)

	Finite Fields- Developer Zone - National Instruments
		An irreducible polynomial, ƒ(X), of degree m is said to be
		Note that an irreducible polynomial is one that cannot be factored to yield lower
		For an irreducible polynomial to be a primitive polynomial,
	zone.ni.com /devzone/cda/ph/p/id/328 (1414 words)

	Root
		Root[f, k] represents the kth root of the polynomial equation f[x] == 0.
		An irreducible polynomial is a polynomial that cannot be factored using integer coefficients.
		In informal usage, the term algebraic number is sometimes used to refer more specifically to roots of irreducible polynomials, or to roots that cannot be represented in terms of radicals.
	support.wolfram.com /mathematica/kernel/Symbols/System/Root.html (163 words)

	irreducible
		irreducible returns TRUE if the polynomial is irreducible over the field implied by its coefficients.
		The polynomial may be either a (multivariate) polynomial over the rationals, a (multivariate) polynomial over a field (such as the residue class ring IntMod(n) with a prime number n) or a univariate polynomial over an algebraic extension (see Dom::AlgebraicExtension).
		Internally, a polynomial expression is converted to a polynomial of type DOM_POLY before irreducibility is tested.
	www.sciface.com /support/doc/40/en/stdlib/irreducible.html (192 words)

	Element Operations
		If the GCD is non-trivial, then this forces a splitting of the defining polynomial, all elements of the field are reduced, and the original element may now be deemed to be zero (it may not be zero because the cofactor of the GCD may be used to perform the simplification).
		Return the minimal polynomial of the element a of the field A, relative to the base field of A. This is the unique minimal-degree irreducible monic polynomial with coefficients in the base field, having a as a root.
		Thus the illusion of a true field is sustained by forcing the minimal polynomial of a to be irreducible, by first performing whatever simplifications of A are necessary for this.
	www.msri.org /about/computing/docs/magma/html/text710.htm (923 words)

	Counting Polynomials over Finite Fields and Analysis of Algorithms
		Elimination of repeated factors replaces a polynomial by square-free ones that contain all the irreducible factors of the original polynomial with exponents reduced to 1.
		is the number of irreducible polynomials of degree n, and a geometrically decaying probability tail.
		A fundamental problem in finite fields is the construction of extension fields, that may be done by using an irreducible polynomial over the ground field with degree equal to the degree of the extension.
	algo.inria.fr /seminars/sem96-97/panario.html (1160 words)

	Number Theory Glossary
		A polynomial which has non-trivial factors is called reducible.
		A polynomial which is not irreducible is called composite.
		A polynomial which has no factors other than 1 is called irreducible.
	www.math.umbc.edu /~campbell/NumbThy/Class/Glossary.html (827 words)

	Eisenstein's irreducibility criterion@Everything2.com
		This result is very useful for producing examples of irreducible polynomials.
		be a nonconstant polynomial with integer coefficients and let p be a prime number.
		By Eisenstein (p=3) we deduce that f(x) is irreducible.
	everything2.com /index.pl?node_id=740620 (489 words)

	Reference.com/Encyclopedia/Irreducible polynomial
		For any field F, the ring of polynomials with coefficients in F is denoted by
		mathbb{Z} of integers, the first two polynomials are reducible, but the other three are irreducible.
		mathbb{Q} of rational numbers, the first three polynomials are reducible, but the other two polynomials are irreducible.
	www.reference.com /browse/wiki/Irreducible_polynomial (735 words)

	Appendix Four
		When n and m are multivariable polynomials, this procedure attempts to answer quickly by substituting each polynomial variable except the highest with a constant.
		The finite field is created by simply being in modular mode over a prime modulus, or by additionally modding out by irreducible polynomials to form a more complex finite field, as described in the section "Polymods." Factoring into irreducibles or square-free polynomials is possible, or polynomials can just be checked for irreducibility.
		Nonetheless, if there are many polynomial variables and the matrix is sparse or has a regular pattern of zeros, expansion by minors can be by far the fastest method.
	www.fordham.edu /lewis/wferm/ap4w.htm (5751 words)

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