Dedekind, Prüfer and Skolem Rings

Working on proofs of Fermat’s Last Theorem, several fascinating mathematical discoveries were made. One famous mistake, which arguably is the reason for the work shown in this thesis, was made by the French mathematician Gabriel Lamé in 1847: when trying to factor Xn + Y n in Z[e2pii/n], Lamé assumed uniqueness of factorisation to prove Fermat’s Last Theorem. This work was shown to be false, though, in a paper by Ernst Kummer which had appeared three years earlier: this factorisation is not unique for n > 23. While trying to save Lamé’s proof, Kummer invented “ideal primes” to restore uniqueness in the factorisation. With this he was able to prove the theorem for more n than Lam´e, in fact, for all prime exponents p such that the class number h(p) of the cyclotomic ring Z[e2pii/p] is not divided by p. In doing so, he – together with Richard Dedekind, who introduced ideals as we know them today – laid the foundation for a whole new part of Algebra: ideal theory. In this thesis, chapter 1 serves as a stepping stone: key concepts of Algebra are introduced and ideals defined. Then, in chapter 2, things become more interesting. Quotient rings and fields are introduced together with fractional ideals which, in turn, enable us to speak of factorisation of ideals. Furthermore, an important example for a ring without unique factorisation is given. In chapter 3 we then delve deeper into unique factorisation: Dedekind domains are introduced, rings in which all ideals can be uniquely factored into products of prime ideals and are multiplicatively invertible. This is then generalised to Prüfer domains where this invertibility needs to be possible for finitely generated ideals only. In between, section 3.4 introduces concepts from Algebraic Number Theory used later in this thesis. Chapter 4 goes away from the general line by explaining the concept of absolute values, places and valuations...