Are there more quadratic reciprocity laws? One can formulate
Legendre and Jacobi symbols in any PID and ask for a quadratic reciprocity
law. The two classic ones are in Z (Gauss-Jacobi) and in the polynomial
ring F_q[t] over a finite field of odd order (Dedekind-Kuhne). It turns out
that there is an especially low-hanging quadratic reciprocity law in the
polynomial ring R[t] over the real numbers. Moreover this law looks
strikingly similar in form to QR in F_q[t] and thus raises the question of
a common generalization. In this talk I will give a generalization to an
nth power reciprocity law in k[t], where k is a perfect field with
procyclic Galois group and containing a primitive nth root of unity.
Moreover I will argue that this is close to being the "natural class of
fields" for which there is a "nice reciprocity law" in k[t]. This is joint
work with Paul Pollack.