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Homeproof of Wedderburn's theorem

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# proof of Wedderburn’s theorem

We want to show that the multiplication operation in a finite division ring is abelian.

We denote the centralizer in $D$ of an element $x$ as $C_{D}(x)$.

$0$ and $1$ are obviously elements of $C_{D}(x)$ and if $y$ and $z$ are, then $x(-y)=-(xy)=-(yx)=(-y)x$, $x(y+z)=xy+xz=yx+zx=(y+z)x$ and $x(yz)=(xy)z=(yx)z=y(xz)=y(zx)=(yz)x$, so $-y,y+z$, and $yz$ are also elements of $C_{D}(x)$. Moreover, for $y\neq 0$, $xy=yx$ implies $y^{{-1}}x=xy^{{-1}}$, so $y^{{-1}}$ is also an element of $C_{D}(x)$.

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

$Z(D)=\bigcap_{{x\in D}}C_{D}(x)$ |

$Z(D)$ is an abelian subring of $D$ and is thus a field. We can consider $D$ and every $C_{D}(x)$ as vector spaces over $Z(D)$ of dimension $n$ and $n_{x}$ respectively. Since $D$ can be viewed as a module over $C_{D}(x)$ we find that $n_{x}$ divides $n$. If we put $q:=|Z(D)|$, we see that $q\geq 2$ since $\{0,1\}\subset Z(D)$, and that $|C_{D}(x)|=q^{{n_{x}}}$ and $|D|=q^{n}$.

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

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

$|D^{*}|=|Z(D^{*})|+\sum_{{x}}[D^{*}:C_{{D^{*}}}(x)]$ |

which gives

$q^{n}-1=q-1+\sum_{{x}}\frac{q^{n}-1}{q^{{n_{x}}}-1}$ |

.

By Zsigmondy’s theorem, there exists a prime $p$ that divides $q^{n}-1$ but doesn’t divide any of the $q^{m}-1$ for $0<m<n$, except in 2 exceptional cases which will be dealt with separately. Such a prime $p$ will divide $q^{n}-1$ and each of the $\frac{q^{n}-1}{q^{{n_{x}}}-1}$. So it will also divide $q-1$ which can only happen if $n=1$.

We now deal with the 2 exceptional cases. In the first case $n$ equals $2$, which would mean $D$ is a vector space of dimension 2 over $Z(D)$, with elements of the form $a+b\alpha$ where $a,b\in Z(D)$. Such elements clearly commute so $D=Z(D)$ which contradicts our assumption that $n=2$. In the second case, $n=6$ and $q=2$. The class equation reduces to $64-1=2-1+\sum_{{x}}\frac{2^{6}-1}{2^{{n_{x}}}-1}$ where $n_{x}$ divides 6. This gives $62=63x+21y+9z$ with $x,y$ and $z$ integers, which is impossible since the right hand side is divisible by 3 and the left hand side isn’t.

## Mathematics Subject Classification

12E15*no label found*

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## Comments

## centralizer link refers to group centralizer definition

Thecentralizer link refers to the group element centralizer definition, not the definition of a centralizer of a ring element.

S. A. G.