We prove the theorem by defining a map , and showing that the kernel of is finite and the image of is a lattice in a hyperplane in . The trickiest part of the proof is showing that the image of spans a hyperplane, and we do this by a clever application of Blichfeld's Lemma 7.1.5.

He is a rather tall, lanky-looking man, with moustache and beard about to turn grey with a somewhat harsh voice and rather deaf. He was unwashed, with his cup of coffee and cigar. One of his failings is forgetting time, he pulls his watch out, finds it past three, and runs out without even finishing the sentence.Koch wrote that:

... important parts of mathematics were influenced by Dirichlet. His proofs characteristically started with surprisingly simple observations, followed by extremely sharp analysis of the remaining problem.I think Koch's observation nicely describes the proof we will give of Theorem 8.1.2.

Units have a simple characterization in terms of their norm.

Let be the number of real and the number of complex conjugate
embeddings of into
, so
.
Define the *log embedding*

for | ||

where is the bounded subset of of elements all of whose coordinates have absolute value at most . Since is a lattice (see Proposition 2.4.5), the intersection is finite, so is finite.

To prove Theorem 8.1.2, it suffices to prove that Im is a lattice in the hyperplane of (8.1.1), which we view as a vector space of dimension .

Define an embedding

given by , where we view via . Thus this is the embedding

Re Im Re Im |

We will use the following lemma in our proof of Theorem 8.1.2.

Since , we have as . It is thus possible to choose the as in the lemma.

Thus suppose . Define a function by

Note that if and only if , so to show that we show that there exists some with .

Let

for | ||

for |

Then is closed, bounded, convex, symmetric with respect to the origin, and of dimension , since is a product of intervals and discs, each of which has these properties. Viewing as a product of intervals and discs, we see that the volume of is

Recall Blichfeldt's Lemma 7.1.5, which asserts that if is a lattice and is closed, bounded, etc., and has volume at least , then contains a nonzero element. To apply this lemma, we take , where is as in (8.1.2). By Lemma 7.1.7, we have . To check the hypothesis of Blichfeld's lemma, note that

Since is nonzero, we also have

Recall that our overall strategy is to use an appropriately chosen to construct a unit such . First, let be representative generators for the finitely many nonzero principal ideals of of norm at most . Since , we have , for some , so there is a unit such that .

Let

In the last step we use (8.1.4).

Let , and note that does not depend on the choice of the ; in fact, it only depends on the field . Moreover, for any choice of the as above, we have

then the fact that would then imply that , which is exactly what we aimed to prove.

If , then we are trying to prove that is a lattice in , which is automatically true, so assume . To finish the proof, we explain how to use Lemma 8.1.9 to choose such that . We have

where and for , and and for , The condition that is that the are not all the same, and in our new coordinates the lemma is equivalent to showing that , subject to the condition that . But this is exactly what Lemma 8.1.9 shows. It is thus possible to find a unit such that . Thus , so , whence , which finishes the proof Theorem 8.1.2.

William Stein 2012-09-24