Fermat's Last Theorem on Regular Prime Exponents

Specialeforsvar ved Hugrún Fjóla Hafsteinsdóttir

Titel: Fermat's Last Theorem on Regular Prime Exponents

Abstract: Fermat's Last Theorem is one of the most celebrated Theorems of Mathematics. The 358 years it took to prove the Theorem were of the outmost significance for Mathematics in general, but especially for the making of a new field, Algebraic Number Theory. Many other advances were made along the way and is it therefore safe to say that it is one of the most influential Mathematical Theorems to this day. The Theorem states that no three positive integers $x,y,z$ exist, that satisfy the equation $x^n+y^n=z^n$ for any integer $n>2$. A prime number $p$ is called regular, if the class number $h$ of the $p$th cyclotomic field $\mathbb{Q}(\zeta_p)$ is not divisible by $p$. Kummer proved the Theorem for regular prime exponents and we go through the proof for regular primes by the traditional way of dividing it up into two cases. Moving on deeper into the property of regularity, we look at the class number $h$ and the class number formula for $\mathbb{Q}(\zeta_p)$ and get a closed formula for $h$. From there we can split $h$ into two factors of natural numbers $h=h_0h^{*}$ and we go on to show that these factors are indeed positive integers. Regularity of primes is here one of the main focuses, but regularity of primes can be expressed in connection to the Bernoulli numbers and we also go into explaining that connection.

Vejleder: Ian Kiming
Censor: Tom Høholdt