Brauer–Nesbitt theorem
In mathematics, the Brauer–Nesbitt theorem can refer to several different theorems proved by Richard Brauer and Cecil J. Nesbitt in the representation theory of finite groups. In modular representation theory, the Brauer–Nesbitt theorem on blocks of defect zero states that a character whose order is divisible by the highest power of a prime "p" dividing the order of a finite group remains irreducible when reduced mod "p" and vanishes on all elements whose order is divisible by "p". Moreover, it belongs to a block of defect zero. A block of defect zero contains only one  ordinary character and only one modular character. Another version states that if "k" is a field of characteristic zero, "A" is a "k"-algebra, "V", "W" are semisimple "A"-modules which are finite dimensional over "k", and Tr = Tr as elements of Hom("A",k), then "V" and "W" are isomorphic as "A"-modules. Let formula_1 be a group and formula_2 be some field. If formula_3 are two finite-dimensional semisimple representations such that the characteristic polynomials of formula_4 and formula_5 coincide for all formula_6, then formula_7 and formula_8 are isomorphic representations. If formula_9 or formula_10, then the condition on the characteristic polynomials can  be changed to the condition that the traces of formula_4 and formula_5 coincide for all formula_6. As a consequence, let formula_14 be a semisimple (continuous) formula_15-adic representations of the absolute Galois group of some field formula_16, unramified outside some finite set of primes formula_17. Then the representation is uniquely determined by the values of the traces of formula_18 for formula_19 (also using the Chebotarev density theorem).