See

Wedderburn's theorem about finite division rings for the
statement of the

theorem. The rather beautiful

proof
here is due to

Witt (from 1931).

Let *F* be our finite division ring. Let *k* be its centre.
Note that *k* is a finite field, as such is has
*q* elements, for some power of a prime number *q*.
Consider *F* as a vector space over *k*, say it has dimension
*n*. It follows that *F* has *q*^{n} elements.
Consider now the multiplicative group *U(F)* of units in *F*.
Because *F* is a division ring, *U(F)* has *q*^{n}-1
elements.

Next we consider the conjugacy classes of the elements of *U(F)*.
An element of *U(F)* in the centre has a singleton conjugacy class.
So consider *a* in *U(F)* that is not central. The centralizer
*C* of *a* in *F* is a subring
of *F* and obviously it contains *k*. Further, since *a*
is not central, *C* is not equal to *F*. It follows from the
argument in the first paragraph that *C* has *q*^{d}
elements. Because *C* is itself a division ring
we have that *q*^{d}-1=|U(C)| is a divisor
of *q*^{n}-1. It follows quickly that
*d* divides *n*.

Now *C(a)* the centralizer of *a* in *U(F)* consists of the nonzero
elements in *C* (and so has *q*^{d} elements) and we
know that the conjugacy class of *a* in *U(F)* has
*|U(F)|/|C(a)|* elements. Since the conjugacy classes of *U(F)*
give a partition of *U(F)* we get a formula

*q*^{n}-1 = q-1 + Sum_{d} (q^{n}-1)/(q^{d}-1) (*)
where the sum is over various proper divisors *d* of *n*
and it could be that some *d*s occur more than once.
(Note that the *q-1* comes from the central elements in *U(F)*.)

Now we will make use of *cyc*_{n}(x) the
*n*th cyclotomic polynomial. It is a factor of any
(*x*^{n}-1)/(x^{d}-1) for a proper divisor *d*
of *n*. So if we evaluate at *q* we get that
*cyc*_{n}(q) divides each
*(q*^{n}-1)/(q^{d}-1). For the same reason it also
divides *q*^{n}-1. So examining (*) we have that
*cyc*_{n}(q) | q-1.

By the way the cyclotomic polynomial is defined
*cyc*_{n}(q) is the product of *(q-e)* where
*e* varies over the primitive *n*th roots of unity
in the complex numbers.

Suppose that *n>1*. We are going to obtain a contradiction from this.
This will show that *n=1* and so *F=k* is a field.
Since *n>1* then *|q-e| > q-1* (draw a picture to see this)
and so we see that *|cyc*_{n}(q)| cannot possibly divide
*q-1*. This contradiction establishes the proof of the theorem.