proof of Wedderburn’s theorem
We want to show that the multiplication operation
in a finite divisionring is abelian
.
We denote the centralizer in of an element as .
Lemma. The centralizer is a subring.
and are obviously elements of and if and are,then , and, so , and are alsoelements of . Moreover, for , implies, so is also an element of .
Now we consider the center of which we’ll call . This isalso a subring and is in fact the intersection of all centralizers.
is an abelian subring of and is thus a field. We can consider and every as vector spaces over of dimension and respectively. Since can be viewed as a module over we find that divides . If we put , we seethat since , and that and .
It suffices to show that to prove that multiplication isabelian, since then and so .
We now consider and apply the conjugacy class formula.
which gives
.
By Zsigmondy’s theorem, there exists a prime that divides but doesn’t divide any of the for , except in 2exceptional cases which will be dealt with separately. Such a prime will divide and each of the . Soit will also divide which can only happen if .
We now deal with the 2 exceptional cases. In the first case equals, which would is a vector space of dimension 2 over, with elements of the form where . Such elements clearly commute so which contradictsour assumption that . In the second case, and . Theclass equation
reduces to where divides 6. This gives with and integers, which is impossible since the right hand side is divisibleby 3 and the left hand side isn’t.