On Indecomposable Definite Hermitian Forms

In this paper, for any given natural numbers n and a, we can construct explicitly position definite indecomposable integral Hermitian forms of rank n over .(.-3) with discriminant a, with the following ten exceptions: n = 2, a = 1, 2, 4, 10; n = 3,a= 1,2,5;n=4,a= 1; n=5,a= 1; and n=7,a= 1. In the exceptional cases there are no forms with the desired properties. The method given here can be applied to solving the problem of constructing indecomposable positive definite Hermitian R.-lattices of any given rank n and discriminant a, where R. is the ring of algebraic integers in an imaginary quadratic field .(.-3) with class number unity.