  Counterparts of several classical results of number theory are proven for the
ring of polynomials with coefficients in a number field. A theorem of Milnor
that determines the Witt ring of a function field is applied to prove an
analogue of Gauss's principal genus theorem for binary quadratic forms with
polynomial coefficients. This is used to help understand when and why quadratic
reciprocity fails in these polynomial rings. Another application is a count of
the number of cyclic subgroups whose order is divisible by four in the primary
decomposition of the torsion subgroup of the Jacobian of certain hyperelliptic
curves. Invariant theory is applied to prove an analogue of a classical theorem
of Fueter to give criteria for an elliptic curve with a polynomial discriminant
and zero $j$-invariant to have no affine points over the associated function
field.
