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 qn elements. Consider now the multiplicative group U(F) of units in F. Because F is a division ring, U(F) has qn-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 qd elements. Because C is itself a division ring we have that qd-1=|U(C)| is a divisor of qn-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 qd 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

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

Now we will make use of cycn(x) the nth cyclotomic polynomial. It is a factor of any (xn-1)/(xd-1) for a proper divisor d of n. So if we evaluate at q we get that cycn(q) divides each (qn-1)/(qd-1). For the same reason it also divides qn-1. So examining (*) we have that cycn(q) | q-1.

By the way the cyclotomic polynomial is defined cycn(q) is the product of (q-e) where e varies over the primitive nth 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 |cycn(q)| cannot possibly divide q-1. This contradiction establishes the proof of the theorem.

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