PYTHAGOREAN TRIPLES AND UNITS IN INTEGRAL GROUP RINGS

In this note, we describe a simple method for finding units of group rings of the form Z[G] = Z[H]#Z[C2] for H an abelian group, and apply this to the case G = D4, the dihedral group of order 8. Here, units may be described as integer points on hyperboloids, and, defining units u, v to be equivalent if they differ by an inner automorphism, we see that this equivalence relation partitions each hyperboloid into finitely many classes.