Kummer's Lemma and Fermat's Last Theorem for Regular Primes – University of Copenhagen

Kummer's Lemma and Fermat's Last Theorem for Regular Primes

Specialeforsvar vd Anna Berger Brusch

Titel: Kummer's Lemma and Fermat's Last Theorem for Regular Primes

Abstract: A prime number $p$ is said to be regular, if the class number $h$ of the $p$th cyclotomic field $\mathbb Q(\zeta_{p})$ is not divisible by $p$. In this thesis, we will prove Kummer's Lemma which enables us to prove Fermat's Last Theorem for regular primes. In order to prove Kummer's Lemma, we need to consider local fields. In particular, we are interested in the field $K_{\mathfrak{p}}$ which is the $\mathfrak{p}$-adic completion of $\mathbb Q(\zeta_{p})$ with respect to the exponential valuation $\nu_{\mathfrak{p}}$ with $\mathfrak{p}=(1-\zeta_{p})$. We consider the class number formula for the field $\mathbb Q(\zeta_{p})$ and find a closed formula for $h$. We will split the class number formula into two factors of natural numbers $h_{0}$ and $h^{*}$ and show that when $p\nmid h^{*}$, the numbers $\log \theta_{k}^{p-1}$ where $\theta_{k}=\frac{\sin k\pi/p}{\sin \pi/p}$ for $k=2,3,...,(p-1)/2$, form a basis for the set of all "real" $\mathfrak{p}$-adic integers with zero trace. Furthermore, it turns out that the property of a prime $p$ being regular is closely connected to the Bernoulli numbers and we will show that a prime number $p\geq 3$ is regular if and only if the numerators of the Bernoulli numbers $B_{2},B_{4},...,B_{p-3}$ are not divisible by $p$.

Vejleder:  Ian Kiming
Censor:   Tom Høholdt, DTU