mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property...
20 KB (2,845 words) - 09:52, 29 April 2024
Eisenstein's criterion (redirect from Eisenstein's Irreducibility Criterion)
Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not...
25 KB (3,592 words) - 03:53, 24 April 2024
product of irreducible monic polynomials. There are efficient algorithms for testing polynomial irreducibility and factoring polynomials over finite...
45 KB (6,162 words) - 21:59, 25 April 2024
mathematics, the concept of irreducibility is used in several ways. A polynomial over a field may be an irreducible polynomial if it cannot be factored over...
4 KB (566 words) - 12:13, 1 April 2021
Algebraically closed field (redirect from Relatively prime polynomials)
only irreducible polynomials in the polynomial ring F[x] are those of degree one. The assertion "the polynomials of degree one are irreducible" is trivially...
12 KB (1,674 words) - 04:51, 4 June 2024
computer algebra the factorization of a polynomial consists of decomposing it into a product of irreducible factors. This decomposition is theoretically...
30 KB (4,620 words) - 19:08, 23 December 2023
polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously, prime polynomials (more correctly, irreducible polynomials)...
59 KB (8,067 words) - 06:41, 26 May 2024
In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor...
29 KB (5,021 words) - 07:02, 15 May 2024
K[X]/(p)} is a field if and only if p is an irreducible polynomial. In fact, if p is irreducible, every nonzero polynomial q of lower degree is coprime with p...
51 KB (8,164 words) - 04:26, 1 April 2024
GF(pm). Because all minimal polynomials are irreducible, all primitive polynomials are also irreducible. A primitive polynomial must have a non-zero constant...
10 KB (1,353 words) - 21:06, 25 May 2024
P(x) is an additive polynomial. Separable polynomials occur frequently in Galois theory. For example, let P be an irreducible polynomial with integer coefficients...
6 KB (783 words) - 21:16, 19 May 2024
characteristic polynomials need not factor according to their roots (in F) alone, in other words they may have irreducible polynomial factors of degree...
11 KB (1,500 words) - 08:26, 28 April 2024
of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the integers as the product of irreducible factors...
28 KB (4,371 words) - 22:04, 1 February 2024
lemma (polynomial) Irreducible polynomial Eisenstein's criterion Primitive polynomial Fundamental theorem of algebra Hurwitz polynomial Polynomial transformation...
5 KB (441 words) - 01:35, 1 December 2023
p(x) is a polynomial, and, for each j, the denominator gj (x) is a power of an irreducible polynomial (that is not factorable into polynomials of positive...
34 KB (7,004 words) - 01:48, 23 January 2024
There is at least one irreducible polynomial for which x is a primitive element. In other words, for a primitive polynomial, the powers of x generate...
26 KB (3,094 words) - 23:14, 21 January 2024
primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has...
23 KB (3,961 words) - 20:33, 7 May 2024
F[X] be an irreducible polynomial and f ' its formal derivative. Then the following are equivalent conditions for the irreducible polynomial f to be separable:...
21 KB (3,073 words) - 22:22, 10 May 2024
Nn be the number of monic irreducible polynomials over F whose degree is equal to n. That is, we are looking at polynomials with coefficients chosen from...
58 KB (8,185 words) - 10:46, 30 May 2024
theory, Hilbert's irreducibility theorem, conceived by David Hilbert in 1892, states that every finite set of irreducible polynomials in a finite number...
4 KB (732 words) - 11:42, 20 August 2021
in Jα. This is the minimal polynomial of α with respect to E/F. It is unique and irreducible over F. If the zero polynomial is the only member of Jα, then...
10 KB (1,447 words) - 22:12, 14 January 2024
{-5}}\right)=9,} but 3 does not divide either of the two factors. Irreducible polynomial Consider p {\displaystyle p} a prime element of R {\displaystyle...
5 KB (707 words) - 19:17, 11 November 2023
algebra and the number of irreducible polynomials over a finite field. The necklace polynomials are a family of polynomials M ( α , n ) {\displaystyle...
8 KB (1,456 words) - 17:16, 25 February 2024
In mathematics, the Conway polynomial Cp,n for the finite field Fpn is a particular irreducible polynomial of degree n over Fp that can be used to define...
18 KB (1,571 words) - 12:05, 6 December 2023
one polynomial (AOP) is a polynomial in which all coefficients are one. Over the finite field of order two, conditions for the AOP to be irreducible are...
4 KB (449 words) - 11:17, 26 June 2023
nonzero element of ( R / m ) {\displaystyle (R/{\mathfrak {m}})} and irreducible polynomials that are monic (that is, their leading coefficients are 1). Hensel's...
42 KB (9,047 words) - 22:44, 1 January 2024
Splitting field (redirect from Splitting field of a polynomial)
^{2}=2\alpha ^{5}+\alpha ^{2}+2\alpha .} Consider the polynomial ring R[x], and the irreducible polynomial x2 + 1. The quotient ring R[x] / (x2 + 1) is given...
17 KB (2,877 words) - 07:00, 18 May 2024
stated as: Every polynomial can be uniquely factorized as the product of its leading coefficient and a product of monic irreducible polynomials. Vieta's formulas...
7 KB (1,159 words) - 12:21, 13 October 2023
− x ( x − 1 ) ( x − 2 ) {\displaystyle y^{2}-x(x-1)(x-2)} is an irreducible polynomial. The ring Z [ x ] / ( x 2 − n ) ≅ Z [ n ] {\displaystyle \mathbb...
20 KB (3,124 words) - 13:53, 23 January 2024
normal extension is an algebraic field extension L/K for which every irreducible polynomial over K that has a root in L splits into linear factors in L. This...
5 KB (940 words) - 14:34, 2 May 2024