# Classical treatment of Fermat’s last Theorem

Specialeforsvar: Marius Enevold Gjesme

Titel: Classical treatment of Fermat’s last Theorem

Adstract: Stated in the 16th century, Fermat’s last Theorem have been of interest in mathematics for a long time. This thesis focuses on the classical approach to solving the problem. We will show the second case of Fermat’s Theorem for regular primes. To do that we have to consider the class number h of the pth cyclotomic field K. We show that h can be written as a product of two natural numbers h0 and h ∗ , which helps us to determine when a prime number p is regular. The prime ideal p = (1 − ζ) in OK is used to consider the completion Kp of K, which we consider in order to establish a regularity criterion for an odd prime p. We obtain that such a prime number p ≥ 3 is regular if and only if p does not divide the numerators of any of the Bernoulli numbers B2, . . . , Bp−3. We furthermore show Kummer’s Lemma, which we then use to finally prove the second case of Fermat’s
last Theorem for regular prime exponents.

Vejleder: Ian Kiming
Censor:   Peter Beelen, DTU