prime factors of xn-1


We list prime factorMathworldPlanetmath of the binomialsMathworldPlanetmath xn-1 in , i.e. in the polynomial ring [x].  The prime factors can always be chosen to be with integer coefficientsMathworldPlanetmath and the number of the prime factors equals to τ(n) (http://planetmath.org/TauFunction); see the proof (http://planetmath.org/FactorsOfNAndXn1).

x-1

x2-1=(x+1)(x-1)

x3-1=(x2+x+1)(x-1)

x4-1=(x2+1)(x+1)(x-1)

x5-1=(x4+x3+x2+x+1)(x-1)

x6-1=(x2+x+1)(x2-x+1)(x+1)(x-1)

x7-1=(x6+x5+x4+x3+x2+x+1)(x-1)

x8-1=(x4+1)(x2+1)(x+1)(x-1)

x9-1=(x6+x3+1)(x2+x+1)(x-1)

x10-1=(x4+x3+x2+1)(x4-x3+x2-x+1)(x+1)(x-1)

x11-1=(x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)(x-1)

x12-1=(x4-x2+1)(x2+x+1)(x2-x+1)(x2+1)(x+1)(x-1)

x13-1=(x12+x11+x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)(x-1)

x14-1=(x6+x5+x4+x3+x2+x+1)(x6-x5+x4-x3+x2-x+1)(x+1)(x-1)

x15-1=(x8-x7+x5-x4+x3-x+1)(x4+x3+x2+x+1)(x2+x+1)(x-1)

x16-1=(x8+1)(x4+1)(x2+1)(x+1)(x-1)

x17-1=(x16+x15+x14++x2+x+1)(x-1)

x18-1=(x6+x3+1)(x6-x3+1)(x2+x+1)(x2-x+1)(x+1)(x-1)

x19-1=(x18+x17+x16++x2+x+1)(x-1)

x20-1=(x8-x6+x4-x2+1)(x4+x3+x2+x+1)(x4-x3+x2-x+1)(x2+1)(x+1)(x-1)

x21-1=(x12-x11+x9-x8+x6-x4+x3-x+1)(x6+x5+x4+x3+x2+x+1)(x2+x+1)(x-1)

x22-1=(x10+x9+x8+x7+x6+x5+x4+x3+x2+x+1)(x10-x9+x8-x7+x6-x5+x4-x3+x2-x+1)(x+1)(x-1)

x23-1=(x22+x21+x20++x2+x+1)(x-1)

x24-1=(x8-x4+1)(x4-x2+1)(x4+1)(x2+x+1)(x2-x+1)(x2+1)(x+1)(x-1)

Note 1.  All factors shown above are irreducible polynomialsMathworldPlanetmath (in the field    of their own coefficients), but of course they (except x±1) may be split into factors of positive degree in certain extension fieldsMathworldPlanetmath; so e.g.

x4+1=(x2+x2+1)(x2-x2+1)inthefield(2).

Note 2.  The 24 examples of factorizations are true also in the fields of characteristic 0, but then many of the factors can be simplified or factored onwards (e.g.  x2+1(x+1)2  if the characteristic (http://planetmath.org/Characteristic) is 2).

Title prime factors of xn-1
Canonical name PrimeFactorsOfXn1
Date of creation 2013-03-22 16:29:51
Last modified on 2013-03-22 16:29:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 12
Author pahio (2872)
Entry type Result
Classification msc 13G05
Related topic GausssLemmaII
Related topic IrreducibilityOfBinomialsWithUnityCoefficients
Related topic FactorsOfNAndXn1
Related topic ExamplesOfCyclotomicPolynomials