Additively Indecomposable Positive Integral Quadratic Forms Original Research Article

Call a positive semidefinite integral quadratic form on a lattice additively indecomposable if it cannot be written as the sum of two such nonzero forms. Criteria for additive indecomposability are proved and quick proofs are given for classical results like the finiteness of the number of equivalence classes of such forms in any dimension and the full classification up to dimension 8 (1-dimensional lattice generated by a norm 1 vector, E6, E7, E8). The same finiteness result is proved for invariant forms on lattices with a finite group acting on them.