# Rational Integer Solutions of a Linear Diophantine Equation of Cyclotomic Integers

Hi everyone.

I am working with lattices in $\mathbb{C}$, and I want to know whether a certain vector is an element of the lattice.

In particular, suppose my lattice vectors are $a$ and $b$ and I want to know whether $c$ (a Gaussian or Eisenstein integer) is in the lattice. The problem reduces to existence of rational integer solutions to the linear diophantine equation $ax+by=c$. Is there any known conditions to determine whether the Gaussian/Eisenstein integer solutions obtained (usual Bezout's) are rational integers?

Any help will be appreciated. Thanks!

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You are probably better off writing everything in terms of a basis over $\mathbb{Z}$, expressing the condition as a matrix equation and using the Smith normal form or another of the standard methods of dealing with integer matrices. –  Felipe Voloch Dec 20 '11 at 10:24
That is precisely what I am trying to avoid, going to the matrix setting. I want to stay in the complex plane for some further generalizations. Thanks anyway! –  Manuel Loquias Dec 20 '11 at 13:06