TITLE: Applications of Character Theory
QUESTION [13 upvotes]: Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character theory).
I would like to know what are some of the other problems which has been solved by the application of character theory especially from number theory and group theory.

REPLY [5 votes]: @Alexander Gruber - A celebrated conjecture of Dedekind asserts that for any finite algebraic extension $K$ of $\mathbb{Q}$, the zeta function $\zeta_K(s)$ is divisible by the Riemann zeta function $\zeta(s)$. That is, the quotient $\zeta_K (s)/\zeta(s)$ is entire. More generally, Dedekind conjectures that if $L$ is a finite extension of $K$, then $\zeta_L(s)/\zeta_K(s)$ should be entire. This conjecture is still open, I believe. By the work of Aramata and Brauer the conjecture is known to be true if $L/K$ is Galois. If $L$ is contained in a solvable extension of $K$, then Uchida and van der Waall have independently proved Dedekind’s conjecture. Proofs rely on properties of M-groups and hence this is a very nice application of character theory to algebraic number theory!