Fermat's Last Theorem For Regular Prime Exponents

Specialeforsvar: Mikkel Bøhlers Nielsen

Titel: Fermat's Last Theorem For Regular Prime Exponents

Resume: We want to prove Fermat's Last Theorem for regular prime exponents. We split the theorem up in two cases, which we prove individually. When working with Fermat's Last Theorem with exponent p, we will primarily be working in the p'th cyclotomic field. First we show some results for the p'th cyclotomic field. We then investigate the ideal class group and define regularity. After this we prove the first case of Fermat's Last Theorem for regular prime exponents. To prove the second case, we start with defining
the p-adic completion of a field, and note some of the properties of the completion. We then define the Bernoulli numbers and state some results about them. After this we find a decomposition of the order of the ideal class group and note some results about the factors of the decomposition. We define the exponential and logarithmic functions on the p-adic numbers. After showing some results about the p-adic completion of the p'th cyclotomic field, we find a basis for the "real" p-adic integers with zero trace. Using this we find a criterion for regularity and proves Kummer's Lemma. Finally we prove the
second case of Fermat's Last Theorem for regular prime exponents.

Vejleder: Ian Kiming
Censor: Tom Høholdt