Note that if we let denote the group of nonzero fractional ideals, then there is an exact sequence

A basic theorem in algebraic number theory is that the class group is finite, which follows from the first part of the following theorem and the fact that there are only finitely many ideals of norm less than a given integer.

Before proving Theorem 10.4.2, we prove a few lemmas. The strategy of the proof will be to start with any nonzero ideal , and prove that there is some nonzero , with very small norm, such that is an integral ideal. Then will be small, since is small. The trick is to determine precisely how small an we can choose subject to the condition that be an integral ideal, i.e., that .

Let be a subset of
. Then is *
if whenever then the line connecting and
lies entirely in . We say that is ** if whenever then also.
If is a lattice in , then the ** of is
the volume of the compact real manifold , which is the
same thing as the absolute value of the determinant of any
matrix whose rows form a basis for .
*

Next assume that . Then for all there is , since . If then the are all in , which is finite since is bounded and is discrete. Hence there exists for arbitrarily small . Since is closed, .

Fix a number field with ring of integers . Let be the embedding

Re Re Im Im |

where are the real embeddings of and are half the complex embeddings of , with one representative of each pair of complex conjugate embeddings. Note that this is

Re Re Im Im

and whose determinant has absolute value equal to the volume
of . By doing the following three column operations,
we obtain a matrix whose rows are exactly the images of
the under - Add times each column with entries Im to the column with entries Re.
- Multiply all columns Im by , thus changing the determinant by .
- Add each columns with entries Re to the the column with entries Im.

Let be any closed, bounded, convex, subset that is symmetric with respect to the origin and has positive volume. Since is closed and bounded,

Suppose is any nonzero fractional ideal of . Our goal is to prove there is an integral ideal with small norm. We will do this by finding an appropriate . By Lemma 10.4.6,

Notice that the right hand side is independent of . It depends only on , , , and our choice of . This completes the proof of the theorem, except for the assertion that can be chosen to give the claim at the end of the theorem, which we leave as an exercise.