The fact that is discrete in implies that has dimension equal to the rank of , as claimed. This last assertion is not obvious, and requires observing that if if a free abelian group that is discrete in a real vector space and , then the rank of equals the dimension of . Here's why this is true. If are a basis for , then has finite index in , since otherwise there would be infinitely many elements of in a fundamental domain for , which would contradict discreteness of . Thus the rank of is , as claimed.
Since is a lattice in , the volume of is finite. Suppose is a basis for . Then if is the matrix whose th row is , then is the volume of . (Take this determinant as the definition of the volume--we won't be using ``volume'' here except in a formal motivating way.)
Let be the ring of integers of . The map is
William Stein 2004-05-06