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单词 ProofOfWedderburnsTheorem
释义

proof of Wedderburn’s theorem


We want to show that the multiplicationPlanetmathPlanetmath operationMathworldPlanetmath in a finite divisionring is abelianMathworldPlanetmath.

We denote the centralizerMathworldPlanetmath in D of an element x as CD(x).

Lemma. The centralizer is a subring.

0 and 1 are obviously elements of CD(x) and if y and z are,then x(-y)=-(xy)=-(yx)=(-y)x, x(y+z)=xy+xz=yx+zx=(y+z)x andx(yz)=(xy)z=(yx)z=y(xz)=y(zx)=(yz)x, so -y,y+z, and yz are alsoelements of CD(x). Moreover, for y0, xy=yx impliesy-1x=xy-1, so y-1 is also an element of CD(x).

Now we consider the center of D which we’ll call Z(D). This isalso a subring and is in fact the intersectionDlmfMathworldPlanetmath of all centralizers.

Z(D)=xDCD(x)

Z(D) is an abelian subring of D and is thus a field. We can considerD and every CD(x) as vector spaces over Z(D) of dimension nand nx respectively. Since D can be viewed as a module overCD(x) we find that nx divides n. If we put q:=|Z(D)|, we seethat q2 since {0,1}Z(D), and that |CD(x)|=qnxand |D|=qn.

It suffices to show that n=1 to prove that multiplication isabelian, since then |Z(D)|=|D| and so Z(D)=D.

We now consider D*:=D-{0} and apply the conjugacy class formula.

|D*|=|Z(D*)|+x[D*:CD*(x)]

which gives

qn-1=q-1+xqn-1qnx-1

.

By Zsigmondy’s theorem, there exists a prime p that divides qn-1but doesn’t divide any of the qm-1 for 0<m<n, except in 2exceptional cases which will be dealt with separately. Such a primep will divide qn-1 and each of the qn-1qnx-1. Soit will also divide q-1 which can only happen if n=1.

We now deal with the 2 exceptional cases. In the first case n equals2, which would D is a vector space of dimension 2 overZ(D), with elements of the form a+bα where a,bZ(D). Such elements clearly commute so D=Z(D) which contradictsour assumptionPlanetmathPlanetmath that n=2. In the second case, n=6 and q=2. Theclass equationMathworldPlanetmath reduces to 64-1=2-1+x26-12nx-1where nx divides 6. This gives 62=63x+21y+9z with x,y and zintegers, which is impossible since the right hand side is divisibleby 3 and the left hand side isn’t.

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更新时间:2025/5/4 5:07:02