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 | ||
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
Re Im Re Im |
We will use the following lemma in our proof of Theorem 8.1.2.
Thus suppose . Define a function by
Let
for | ||
for |
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
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
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
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
William Stein 2012-09-24