It is shown that the solvability of certain quadratic forms is necessary for the existence of
some generalized Hadamard matrices. The number-theoretic consequences of this are
explored. In particular, non-existence results for generalized “Butson” Hadamard matrices
BH (h, pf) and BH (h, 2pf) are given, where p is a prime of the form 4k+ 3.

The determinant of a generalized Hadamard matrix over its group ring factored out by the
relation Σ gϵG g= 0 is shown to have certain number theoretic properties. These are
exploited to prove the non-existence of many generalised Hadamard matrices for groups
whose orders are divisible by 3, 5 or 7. For example the GH (15, C 15), GH (15, C 3) and GH
(15, C 5) do not exist. Also for certain n and G we find the set of determinants of the GH (n, G)
matrices.