power of a prime. Proofs The proof of Brauer's induction theorem exploits the ring structure of Char(G) (most proofs also make use of a slightly larger ring, Char*(G), which consists of -combinations of irreducible characters, where ω is a primitive complex |G|-th root of unity). The set of integer combinations of characters induced from linear characters of Brauer elementary subgroups is an ideal I(G) of Char(G), so the proof reduces to showing that the trivial character is in I(G). Several proofs of the theorem, beginning with a proof due to Brauer and John Tate, show that the trivial character is in the analogously defined ideal I*(G) of Char*(G) by concentrating attention on one prime p at a time, and constructing integer-valued elements of I*(G) which differ (elementwise) from the trivial character by (integer multiples of) a sufficiently high power of p. Once this is achieved for every prime divisor of |G|, some manipulations with congruences and algebraic integers, again exploiting the fact that I*(G) is an ideal of Ch*(G), place the trivial character in I(G). An auxiliary result here is that a -valued class function lies in the ideal I*(G) if its values are all divisible (in ) by |G|. Brauer's induction theorem was proved in 1946, and there are now many alternative proofs. In 1986, Victor Snaith gave a proof by a radically different approach, topological in nature (an application of the Lefschetz fixed-point theorem). There has been related recent work on the question of finding natural and explicit forms of Brauer's theorem, notably by Robert Boltje. Applications Using Frobenius reciprocity, Brauer's induction theorem leads easily to his fundamental characterization of characters, which asserts that a complex-valued class function of G is a virtual character if and only if its restriction to each Brauer elementary subgroup of G is a virtual character. This result, together with the fact that a virtual character θ is an irreducible character if and only if θ(1) > 0 and (where is the usual inner product on the ring of complex-valued class functions) gives a means of constructing irreducible characters without explicitly constructing the associated representations. An initial motivation for Brauer's induction theorem was application to Artin L-functions. It shows that those are built up from Dirichlet L-functions, or more general Hecke L-functions. Highly significant for that application is whether each character of G is a non-negative integer combination of characters induced from linear