irreducible polynomial

Let  f(x)=a0+a1x++anxn  be a polynomialMathworldPlanetmathPlanetmathPlanetmath with complex coefficients aν and with the degree (  n>0.  If f(x) can not be written as product of two polynomials with positive degrees and with coefficients in the field  (a0,a1,,an),  then the polynomial f(x) is said to be .  Otherwise, f(x) is reducible.

Examples.  All linear polynomials are .  The polynomials x2-3, x2+1 and x2-i are (although they split in linear factors in the fields (3), (i) and (1+i2), respectively).  The polynomials x4+4 and x6+1 are not .

The above definition of polynomial is special case of the more general setting where f(x) is a non-constant polynomial in the polynomial ring K[x] of a field K; if f(x) is not expressible as product of two polynomials with positive degrees in the ring K[x], then f(x) is (in K[x]).

Example.  If K is the Galois field with two elements (0 and 1), then the trinomial x2+x+1 of K[x] is (because an equation  x2+x+1=(x+a)(x+b)  would imply the two conflicting conditions  a+b=1  and  ab=1).

Title irreducible polynomial
Canonical name IrreduciblePolynomial
Date of creation 2013-03-22 14:24:22
Last modified on 2013-03-22 14:24:22
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 18
Author pahio (2872)
Entry type Definition
Classification msc 12D10
Synonym prime polynomial
Synonym indivisible polynomial
Related topic EisensteinCriterion
Related topic Irreducible
Related topic Monic2
Defines irreducible polynomial
Defines reducible