Irreducible Polynomial


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irreducible polynomial

[‚ir·ə′dü·sə·bəl ‚päl·ə′nō·mē·əl]
(mathematics)
A polynomial is irreducible over a field K if it cannot be written as the product of two polynomials of lesser degree whose coefficients come from K. Also known as irreducible function.

Irreducible Polynomial

 

a polynomial that cannot be factored into factors of lower degree. The possibility of factoring a polynomial into factors and the irreducibility property depend on the numbers that can be coefficients of the polynomial. Thus, the polynomial x3 + 2 is irreducible if only rational numbers are admitted as coefficients; but it is factored into the product of two irreducible polynomials

if any real numbers can be taken as the coefficients and into the product of three factors

if complex numbers are the coefficients. In the general case, the concept of irreducibility is defined for polynomials with coefficients belonging to an arbitrary field. A polynomial with rational coefficients that cannot be factored into factors of lower degree with rational coefficients is often called an irreducible polynomial.

REFERENCE

Kurosh, A. G. Kurs vysshei algebry, 9th ed. Moscow, 1968.
References in periodicals archive ?
An irreducible polynomial is polynomial that cannot be written as a product of nontrivial polynomials over the same field.
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This analysis is based on the study of the mathematical model of polynomial representation, in which it is stated: If p(x) is the irreducible polynomial, then the multiplication of two elements of the field, represented as the polynomials A(x) and B(x) is the algebraic product of the two polynomials, and the modulus operation of the polynomial p(x), also known as modular reduction, is that shown in equation 1.
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Let [alpha] be a root of an irreducible polynomial g(x) of degree n over GF(2).
Therefore we use locator set [mathematical expression not reproducible], where [mathematical expression not reproducible] is formal derivative of denominator [mathematical expression not reproducible] and [mathematical expression not reproducible] is an irreducible polynomial on [mathematical expression not reproducible].
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(1) Let n = 5k, for k odd, and in GF([2.sup.5]), let [x.sup.5] + [x.sup.2] + 1 = 0, be the irreducible polynomial. If e = 1 + x + [x.sup.3], b = [x.sup.2], c = [x.sup.3] then the semifield (S, +, *) of order [2.sup.5k] admits a subfield isomorphic to GF(4).