Fermat's last theorem and undecidability - useful books
|This page lists books related to the article Is Fermat's last theorem undecidable?|
General Books - Fermat's Last TheoremFermat's Last Theorem: The Story of a Riddle That Confounded the World's Greatest Minds for 358 Years by Simon Singh combines a history of the mathematics leading up to the proof with a biographical description of Wiles' work - how he decided he was going to prove it when he was 10 years old, and succeeded 30 years later. Note that this book is essentially the same as the earlier Fermat's Enigma : The Epic Quest to Solve the World's Greatest Mathematical Problem
Alternatively there's Fermat's Last Theorem : Unlocking the Secret of an Ancient Mathematical Problem by AMIR D. ACZEL this is shorter (160 pages compared with 362 for Singh's book), but covers a wider area of the history of mathematics, and so is less detailed. It also has more on the history of Modular Forms.
You may also be interested in the books by Charles J. Mozzochi. The Fermat Diary, gives a description of Wiles' work on the proof, and The Fermat Proof which is a very short book (only 50 pages in total) gives an outline of the mathematics used in the proof.
General Books - Undecidability
If you want to know about undecidability in mathematics, there are a couple of books, both of which deal with the philosophical developments in a readable way, although they both could also be described as being about the history of mathematics. In fact Pi in the Sky by John Barrow is really two different books in one, the first part being a history of counting, the second a description of the ideas in mathematical logic which arose around the ideas of Cantor and Godel. Unfortunately Barrow is a believer in 'real' undecidability, which I have argued against, but if you overlook that the book is well worth reading. The other book is Mathematics, The loss of certainty by Morris Kline, who puts forward the view that mathematics has an image of certainty, but in truth has never deserved this. The trouble is he seems to claim much more than the evidence supports, and I found the early part of the book rather off-putting. However it does improve in the later chapters.
More advanced books - Fermat's last theoremI've heard that to understand Wiles' proof of Fermat's Last Theorem would take ten years of continuous study from the level of undergraduate mathematics. I'm not sure I believe it, so don't let it discourage you. If you're looking for an informal description of the route to the proof then you might like Notes on Fermat's Last Theorem by Alfred van der Poorten. However be warned - informal here means 'in the style of mathematicians talking to each other', so you need fairly advanced mathematics as a prerequisite and you also need to be happy with this 'informal' style.
Possibly more suited to the mathematics student wanting to understand some of the proof is Algebraic number theory and Fermat's last theorem by Ian Stewart and David Tall. As the title suggests this is mostly about Algebraic number theory, but it does include some material on modular forms and elliptic curves, which are a crucial part of the proof. For myself, I'm working my way through Invitation to the Mathematics of Fermat-Wiles by Yves Hellegouarch, which seems to me to be the best choice for getting a real grasp of the proof.