  The study of regular incidence structures such as projective planes and
symmetric block designs is a well established topic in discrete mathematics.
Work of Bruck, Ryser and Chowla in the mid-twentieth century applied the
Hasse-Minkowski local-global theory for quadratic forms to derive non-existence
results for certain design parameters. Several combinatorialists have provided
alternative proofs of this result, replacing conceptual arguments with
algorithmic ones. In this paper, we show that the methods required are purely
linear-algebraic in nature and are no more difficult conceptually than the
theory of the Jordan Canonical Form. Computationally, they are rather easier.
We conclude with some classical and recent applications to design theory,
including a novel application to the decomposition of incidence matrices of
symmetric designs.
