Wedderburn's little theorem
In mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skew-fields and fields. 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 Herbland quotient vanishes by finiteness, coincides with , which in turn vanishes by Hilbert 90.
The Artin–Zorn theorem generalizes the theorem to alternative rings.
History
The original proof was given by Joseph Wedderburn in 1905,[1] 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 (Parshall 1983), Wedderburn's first proof was incorrect – it had a gap – and his subsequent proofs came 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.[1] Witt's proof is sketched below. Alternatively, the theorem is a consequence of the Skolem–Noether theorem.
Sketch of proof
Let be a finite domain. For each nonzero , the map
is injective; thus, surjective. Hence, has a left inverse. By the same argument, has a right inverse. A is thus a skew-field. Since the center of is a field, is a vector space over with finite dimension n. Our objective is then to show . If is the order of , then A has order . For each that is not in the center, the centralizer of x has order where d divides n. Viewing , and as groups under multiplication, we can write the class equation
where the sum is taken over all representatives that is not in and d are the numbers discussed above. and both admit factorization in terms of cyclotomic polynomials . After cancellation, we see that divides and , so it must divide . So we reach contradiction unless .
Notes
- ^ a b Lam (2001), Template:Google books quote
References
- Parshall, K. H. (1983). "In pursuit of the finite division algebra theorem and beyond: Joseph H M Wedderburn, Leonard Dickson, and Oswald Veblen". Archives of International History of Science. 33: 274–99.
- Lam, Tsit-Yuen (2001). A first course in noncommutative rings. Graduate texts in mathematics. Vol. 131 (2 ed.). Springer. ISBN 0-387-95183-0.