There is an algebraic K-theory seminar at my school and we are struggling to find applications from areas other than topology. We'd like a nice statement like "If X then Y" whose proof makes unexpected use of $K_0(R)$ or $K_1(R)$, where X and Y are purely algebraic statements about rings (or something like that).
An example from group cohomology (not K-theory) is the following Sylow-style theorem, which is elementary to state.
Problem A. Suppose $G$ is a finite group of order $mn$ where $\gcd(m,n)=1$. Suppose $A$ is a normal abelian subgroup of order $m$. Prove that $G$ has a subgroup of order $n$, and show that any two such subgroups are conjugate.
Surprisingly a quick proof can be given with group cohomology. We're looking for something similar, except maybe for rings/fields, and instead of group cohomology it must use K-theory.
I've also seen the following result as a bonus problem in commutative algebra, and my prof told me the only solution he knew used K-theory. I don't know the solution though.
Problem B. Suppose $R\subseteq S$ is an integral extension of commutative domains. Suppose that $S$ is a PID. Prove that $R$ is a PID.
Not sure if the above statement is true as it stands, but I'm
definitely pretty sure it's true if you assume that $R$ is a finitely-generated free $R$-module.
Anything else like Problem B above?