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单词 IrreduciblePolynomial
释义

irreducible polynomial


Let  f(x)=a0+a1x++anxn  be a polynomialMathworldPlanetmathPlanetmathPlanetmath with complex coefficients aν and with the degree (http://planetmath.org/Polynomial)  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).

Titleirreducible polynomial
Canonical nameIrreduciblePolynomial
Date of creation2013-03-22 14:24:22
Last modified on2013-03-22 14:24:22
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id18
Authorpahio (2872)
Entry typeDefinition
Classificationmsc 12D10
Synonymprime polynomial
Synonymindivisible polynomial
Related topicEisensteinCriterion
Related topicIrreducible
Related topicMonic2
Definesirreducible polynomial
Definesreducible
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更新时间:2025/5/3 14:10:01