Tittel: Dedakind zeta functions
Veileder: Petter Andreas Bergh
Sammendrag: In 1847 the French mathematician Gabriel Lamé proposed a proof of Fermat’s Last Theorem to the Paris Academy. However the theorem was not established until recently by Wiles et.al. So Lamé was wrong. His blunder was to assume that the cyclotomic integers $\mathbb{Z}[\zeta_p]$ had unique factorization for all primes $p$, something Kummer had proven was not true even before Lamé proposed his proof.
The ring $\mathbb{Z}[\zeta_p]$ is the ring of integers of the cyclotomic extension $\mathbb{W}(\zeta_p)$ of $\mathbb{Q}$. To any number field $K$, which we will very soon define, we can associate a ring of integers $\mathcal{O}_K$ which mimics how $\mathbb{Z}$ lies in $\mathbb{Q}$. The failure of unique factorization in such rings is measured by what is called the ideal class group. Towards the end of this thesis we will give a formula for calculating the class number - the order of the ideal class group. Although such a ring of integers may lack unique factorization, they have the amazing property that every non-zero ideal factorize uniquely into prime ideals. At the heart of this thesis is the study of $\mathcal{O}_K$.