Catalan's Conjecture by René Schoof

Author:René Schoof
Language: eng
Format: epub, pdf
Publisher: Springer London, London

Theorem 9.5

Let p be a prime and let be a prime ideal of the ring of degree 1. Then is principal for every element θ of the Stickelberger ideal.

Fig. 9.1Ernst Eduard Kummer (1810–1893)

Proof

Set and let . As before, we write , where . Let be a prime ideal of of degree 1. Its residue field is , where l is a prime congruent to . Set . The restriction maps induce an isomorphism

We let G act on by setting . Similarly, we write

where , and we extend the action of δ to by setting . We have and .

We fix a primitive lth root of unity ζ l and a nontrivial character χ from to the group of pth roots of unity μ p in . Then we define the Gaussian sum τ by

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.