second proof of Wedderburn’s theorem

We can prove Wedderburn’s theorem,without using Zsigmondy’s theorem on the conjugacy class formula of the first proof; let Gn set of n-th roots of unityMathworldPlanetmath and Pn set of n-th primitive roots of unity and Φd(q) the d-th cyclotomic polynomialMathworldPlanetmath.
It results

  • Φn(q)=ξPn(q-ξ)

  • p(q)=qn-1=ξGn(q-ξ)=dnΦd(q)

  • Φn(q)[q], it has multiplicative identityPlanetmathPlanetmath and Φn(q)qn-1

  • Φn(q)qn-1qd-1with dn,d<n

by conjugacy class formula, we have:


by last two previous properties, it results:


because Φn(q) divides the left and each addend of xqn-1qnx-1 of the right member of the conjugacy class formula.
By third property


If, for n>1,we have |Φn(q)|>q-1, then n=1 and the theorem is proved.
We know that


by the triangle inequality in


as ξ is a primitive root of unity, besides




therefore, we have

Title second proof of Wedderburn’s theorem
Canonical name SecondProofOfWedderburnsTheorem
Date of creation 2013-03-22 13:34:39
Last modified on 2013-03-22 13:34:39
Owner Mathprof (13753)
Last modified by Mathprof (13753)
Numerical id 17
Author Mathprof (13753)
Entry type Proof
Classification msc 12E15