DescriptionOne of the most famous results in mathematics is Fermat's Last Theorem, which says that the equation $$ x^n+y^n=z^n $$ has no solutions in non-zero integers $x,y,z$ if $n>2$ is an integer. This was a famous open problem for over 350 years until Andrew Wiles (with some help from Richard Taylor) proved it around 1994. Before Wiles's proof, the best general result was due to E. Kummer, and says that the above equation has no solutions when $n$ is a so-called regular prime. As far as we know, regular primes are fairly common. For example, all primes less than 100 are regular except for 37, 59 and 67. It has been conjectured that around 60.65% of all prime numbers are regular (in a certain asymptotic sense), but this has not been proved so far, and we don't even know for sure whether there are infinitely many regular primes! In this project we will define what it means for a prime to be regular, and work our way through the proof of Fermat's Last Theorem for regular primes. On the way, we will encounter several notions from algebraic number theory such as number fields generated by roots of unity (cyclotomic fields), prime ideals, class groups and unique factorisation. We will initially follow the steps in K. Conrad's notes (see below) and introduce the necessary concepts from algebraic number theory as we go along. For further individual studies in Epiphany term, students are free to choose any direction that naturally carries on from the Michaelmas material, for example, any of the later chapters in Ribenboim's book would be suitable.
PrerequisitesAlgebra II is necessary. It is strongly recommended that you take Number Theory IV in parallel with this project. Note that it is not necessary to take NT IV if you want to do this project, but it will be helpful. If you haven't taken ENT II, you will need to read up on Fermat's proof of FLT for $n=4$, using Pythagorean triples.Resources
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email: Alexander Stasinski