Cyclotomic Fields and Fermat's Last Theorem

Abstract

Number theory is in a sense among the oldest disciplines of mathematics. A running theme is that many number theoretic problems are very easy to state and understand. However, actually solving these problems often requires much work or some very clever trick. An extreme case of this would be Fermat's last theorem: originally a conjecture by French lawyer Pierre de Fermat whose statement is understandable to anyone who knows how to add and take powers; however, it took over three and a half centuries to prove, despite many of history's greatest mathematicians giving it their best efforts.

During the $17^{\textup{th}}$ century Fermat made a note in a copy of Diophantus's Arithmetica of a statement for which he famously claimed to have a marvelous proof; this is the very same note where he famously wrote that the margins of the book were too small for said proof. The statement he wrote down was simply that the equation \[ x^n + y^n =z^n \] has no solutions such that $x$, $y$ and $z$ are all non-zero integers when $n$ is an integer strictly greater than 2. This is the statement that would eventually became known as Fermat's last conjecture. For centuries this conjecture remained uncracked until the matter was finally settled by Andrew Wiles toward the end of the $20^\textup{th}$ century. Wiles managed to prove that the conjecture was in fact true, earning it the title it is currently known by: Fermat's last theorem.

Before yielding to Wiles, Fermat's last theorem was highly sought after. Many a great mathematician attempted to prove the statement long before Wiles, including the likes of Euler and Gauss, but they were only ever able to establish special cases. Among these was French mathematician Gabriel Lamé. In fact, Lamé was convinced that he had essentially found a proof for the statement \cite{edwa}. The trick was, according to Lamé, to look at a certain factorization of $x^n + y^n$ as a \textit{cyclotomic integer}. He thought that he could use the uniqueness of this factorization to prove Fermat's last theorem. However, his proof turned out to have a fatal flaw.

Cyclotomic integers are essentially a collection of complex numbers with an arithmetic very similar to that of the integers. For instance, much like integers, cyclotomic integers have irreducible numbers that all cyclotomic integers factor into. Here irreducible essentially means that there are no ``meaningful'' cyclotomic integers dividing the factor and that the number itself is ``meaningful'' as a divisor; some cyclotomic integers divide all other cyclotomic and in this sense behave very much like $\pm 1$, since knowing that they divide a given number tells us nothing about that number. The collection of cyclotomic integers can be different depending on what specific cyclotomic integers we are talking about. Each collection of cyclotomic integers corresponds to the complex number $\omega = e^{\frac{2 \pi i}{m}}$ for some integer $m$, and can be constructed by extending the integers to also include $\omega$ and all possible finite sums and products of integers and $\omega$, with repetition of numbers allowed.

On the $4^\textup{th}$ of January 1847, Lamé proposed to use the cyclotomic integers corresponding to $n$ to factorize $x^n + y^n$ . Where he went wrong was in assuming that factorization into irreducible cyclotomic integers was in fact unique. That this was not always the case had in fact been proven in a paper by German mathematician Ernst Kummer three years prior to Lamé presenting his supposed proof \cite{edwa}. One integer $n$ such that unique factorization fails for the corresponding cycltomic integers is $n=23$, as is shown in this paper.

While Kummer's paper was a death-blow to Lamé's arguments, Kummer was convinced that unique factorization into irreducible numbers was in fact valid for cyclotomic integers if one also considered what he called \textit{ideal numbers}. Using these he was able to use Lamé's factorization to prove Fermat's last theorem for special primes that are known as \textit{regular primes}, although his proof was much more complicated than that which Lamé had proposed. A proof of Fermat's last theorem for regular primes can be found in the last section of this paper.

Popular Abstract

The two main aims of this paper are to show that there are rings of cyclotomic rings which are not UFD's and to prove Fermat's last theorem for regular primes, assuming the statement of Kummer's lemma holds.

• author: Runyeon Odeberg, Samuel
• supervisor: Arne Meurman
• organization: Mathematics (Faculty of Engineering); Mathematics (Faculty of Sciences)
• course: MATK11 20211
• year: 2021
• type: M2 - Bachelor Degree
• subject: Mathematics and Statistics
• keywords: pure mathematics, number theory, algebra, algebraic number theory, cyclotomic fields, cyclotomic rings, cycltomic integers, Fermat's last theorem
• publication/series: Bachelor's Theses in Mathematical Sciences
• report number: LUNFMA-4114-2021
• ISSN: 1654-6229
• other publication id: 2021:K17
• language: English
• id: 9054900
• date added to LUP: 2021-09-24 15:53:30
• date last changed: 2021-11-01 18:47:02