In mathematics, Wedderburn's little theorem states that every finite division ring is a field; thus, every finite domain is a field.
In other words, for finite rings, there is no distinction between domains, division rings and fields.
The Artin–Zorn theorem generalizes the theorem to alternative rings: every finite alternative division ring is a field.
[1] The original proof was given by Joseph Wedderburn in 1905,[2] who went on to prove the theorem in two other ways.
Another proof was given by Leonard Eugene Dickson shortly after Wedderburn's original proof, and Dickson acknowledged Wedderburn's priority.
However, as noted in (Parshall 1983), Wedderburn's first proof was incorrect – it had a gap – and his subsequent proofs appeared only after he had read Dickson's correct proof.
On this basis, Parshall argues that Dickson should be credited with the first correct proof.
A simplified version of the proof was later given by Ernst Witt.
be a finite division algebra with center
elements; so they are isomorphic and thus are conjugate by Skolem–Noether.
in our case) cannot be a union of conjugates of a proper subgroup; hence,
A later "group-theoretic" proof was given by Ted Kaczynski in 1964.
[4] This proof, Kaczynski's first published piece of mathematical writing, was a short, two-page note which also acknowledged the earlier historical proofs.
The theorem is essentially equivalent to saying that the Brauer group of a finite field is trivial.
In fact, this characterization immediately yields a proof of the theorem as follows: let K be a finite field.
Since the Herbrand quotient vanishes by finiteness,
, which in turn vanishes by Hilbert 90.
The triviality of the Brauer group can also be obtained by direct computation, as follows.
is a cyclic group of order
and the standard method of computing cohomology of finite cyclic groups shows that
For each nonzero x in A, the two maps are injective by the cancellation property, and thus, surjective by counting.
It follows from elementary group theory[5] that the nonzero elements of
form a group under multiplication.
contains the distinct elements
as groups under multiplication, we can write the class equation where the sum is taken over the conjugacy classes not contained within
are defined so that for each conjugacy class, the order of
, and therefore by taking the norms, To see that this forces
using factorization over the complex numbers.
-th roots of unity, set
and then take absolute values For