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 set of n-th roots of unity and set of n-th primitive
roots of unity and the d-th cyclotomic polynomial.
, it has multiplicative identity and
by conjugacy class formula, we have:
by last two previous properties, it results:
divides the left and each addend of
of the right member of the conjugacy class formula.
By third property
If, for ,we have , then and the theorem is proved.
We know that
by the triangle inequality in
as is a primitive root of unity, besides
therefore, we have
|Title||second proof of Wedderburn’s theorem|
|Date of creation||2013-03-22 13:34:39|
|Last modified on||2013-03-22 13:34:39|
|Last modified by||Mathprof (13753)|