Integral invariants for vectors over local fields

This paper considers isometric invariants of vectors in lattices (quadratic forms) over the ring of integers in a local field for the prime 2. By extending the notion of order to vectors in the lattice we obtain a set of invariants which enable the general vector to be decomposed into a sum of simple vectors.The lengths of these simple vectors are invariant modulo certain powers of 2 and these lengths together with the original invariants form a complete set for the 2-adic integers.In the special case where there are no one dimensional, orthogonal sublattices (improper quadratic forms) the invariants form a complete set for all local fields.Let F be a local field, that is a field complete with respect to a discrete, non-archimedean valuation with a finite residue class field, and R the ring of integers in F. Denote by v(x) the order of the