Wedderburn's little theorem

In mathematics, Wedderburn's little theorem states that every finite division ring 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. The original proof was given by Joseph Wedderburn in 1905, who went on to prove it 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 , 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. Witt's proof is sketched below. Alternatively, the theorem is a consequence of the Skolem–Noether theorem by the following argument. Let be a finite division algebra with center . Let and denote the cardinality of . Every maximal subfield of has elements; so they are isomorphic and thus are conjugate by Skolem–Noether. But a finite group (the multiplicative group of 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. 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, coincides with , which in turn vanishes by Hilbert 90. Let A be a finite domain. For each nonzero x in A, the two maps are injective by the cancellation property, and thus, surjective by counting.