Definite quadratic forms over

Let R be a principal ideal domain with quotient field F. An R-lattice is a free R-module of finite rank spanning an inner product space over F. The classification problem asks for a reasonably effective set of criteria to determine when two given R-lattices are isometric; that is, when there is an inner-product preserving isomorphism carrying one lattice onto the other. In this paper R is the polynomial ring , where is a finite field of odd order q. For -lattices as for -lattices the theory splits into “definite” and “indefinite” cases, and this paper settles the classification problem in the definite case.