History[edit]

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.^[2] Witt's proof is sketched below. Alternatively, the theorem is a consequence of the Skolem–Noether theorem by the following argument.^[3] Let ${\displaystyle D}$ be a finite division algebra with center ${\displaystyle k}$. Let ${\displaystyle [D:k]=n^{2}}$ and ${\displaystyle q}$ denote the cardinality of ${\displaystyle k}$. Every maximal subfield of ${\displaystyle D}$ has ${\displaystyle q^{n}}$ elements; so they are isomorphic and thus are conjugate by Skolem–Noether. But a finite group (the multiplicative group of ${\displaystyle D}$ in our case) cannot be a union of conjugates of a proper subgroup; hence, ${\displaystyle n=1}$.


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.