What is the significance of rings in understanding Fermat's Last Theorem?

[lineintegral1]

Hello all,

I'm a freshman undergraduate (UCLA) and have finished my study lower division calculus (multivar, diff eq, etc.). I am currently in linear algebra. I will be taking analysis two quarters from now. Lately, I have been doing research on Fermat's Last Theorem. Can anyone here summarize the proof or give me a general idea as to how it works? I understand it is very long; I'd just like a brief overview. I also understand that the topic of rings is very important. What is a ring and how do rings apply to the proof?

Greatly appreciated!

 

[Robert1986]

Hello all,

I'm a freshman undergraduate (UCLA) and have finished my study lower division calculus (multivar, diff eq, etc.). I am currently in linear algebra. I will be taking analysis two quarters from now. Lately, I have been doing research on Fermat's Last Theorem. Can anyone here summarize the proof or give me a general idea as to how it works? I understand it is very long; I'd just like a brief overview. I also understand that the topic of rings is very important. What is a ring and how do rings apply to the proof?

Greatly appreciated!

Hi, I'm also an undergrad. FLT is a very difficult thing to prove. I have no idea how it works, but it uses things called epliptic curves.

From what I understand, rings are very important to the theorem. A ring is a set with two operations defined for the set and these operations meet certain axioms. Look up Ring Theory in wikipedia for a better look.

There is a book called Fermat's Last Theorem For Amateurs that is a good read.

 

[brydustin]

Hello all,

I'm a freshman undergraduate (UCLA) and have finished my study lower division calculus (multivar, diff eq, etc.). I am currently in linear algebra. I will be taking analysis two quarters from now. Lately, I have been doing research on Fermat's Last Theorem. Can anyone here summarize the proof or give me a general idea as to how it works? I understand it is very long; I'd just like a brief overview. I also understand that the topic of rings is very important. What is a ring and how do rings apply to the proof?

Greatly appreciated!

Don't fool yourself. Study algebra for at least 6 more years, then ask the question again. Have you taken group theory, fields, or rings course?
I don't understand why there are so many posts on FLT on PF. Every month you also find some nut-job that claims to have the proof using only simple algebra (as was Fermat's original FALSE claim). Its not something that is worth even discussing unless you have studied at least a year or two of graduate level algebra, no offense. Its just that the tools used to solve FTL are much much more complicated than the result itself. Study something that you can actually get something out of ... and come back to FLT in a few years, its seductive but also a waste of time if you don't have the tools. Remember A.W. spent years of his life solving it, and it practically destroyed him.

 

[Robert1986]

Don't fool yourself. Study algebra for at least 6 more years, then ask the question again. Have you taken group theory, fields, or rings course?
I don't understand why there are so many posts on FLT on PF. Every month you also find some nut-job that claims to have the proof using only simple algebra (as was Fermat's original FALSE claim). Its not something that is worth even discussing unless you have studied at least a year or two of graduate level algebra, no offense. Its just that the tools used to solve FTL are much much more complicated than the result itself. Study something that you can actually get something out of ... and come back to FLT in a few years, its seductive but also a waste of time if you don't have the tools. Remember A.W. spent years of his life solving it, and it practically destroyed him.

So, I have suspected this myself, but are most people in Math in agreement that Fermat didn't really prove it? I know he couldn't have used the algebra Wiles did, but do most mathematicians think that he never really proved it? I had a professor explain how he thought that Fermat might have proven it using Fermat's infinite descent method. He gave a "proof" of FLT using the infinite descent method, then pointed out where he had made a fatal flaw in the argument.

 

[Landau]

So, I have suspected this myself, but are most people in Math in agreement that Fermat didn't really prove it?

Fermat wrote

I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.

Decide for yourself whether you believe this. Wikipedia says:

The mathematical techniques used in Fermat's "marvellous" proof are unknown. Only one detailed proof of Fermat has survived, the above proof that no three coprime integers (x, y, z) satisfy the equation x4 − y4 = z2.

Taylor and Wiles's proof relies on mathematical techniques developed in the twentieth century, which would be alien to mathematicians who had worked on Fermat's Last Theorem even a century earlier. Fermat's alleged "marvellous proof", by comparison, would have had to be elementary, given mathematical knowledge of the time, and so could not have been the same as Wiles' proof. Most mathematicians and science historians doubt that Fermat had a valid proof of his theorem for all exponents n.

Harvey Friedman's grand conjecture implies that Fermat's last theorem can be proved in elementary arithmetic, a rather weak form of arithmetic with addition, multiplication, exponentiation, and a limited form of induction for formulas with bounded quantifiers. [116] Any such proof would be elementary but possibly too long to write down.

 

[Islam Hassan]

Do we know of any collaborative or individual effort to try to prove FLT using only the mathematics known to Fermat at the time? Would any academic institution fund such undertaking or is it considered too whacky?

IH

 

[brydustin]

Do we know of any collaborative or individual effort to try to prove FLT using only the mathematics known to Fermat at the time? Would any academic institution fund such undertaking or is it considered too whacky?

IH

Hi, turns out that there are a few people who believe it can be done using simple mathematics. When I first made a reply to this post months back I was in a history of math class and this was my topic; I remember there are a few that believe it can be done. But to be honest, they probably don't devote a lot of time to it. For them its a bit like an emotional feeling. As far as I know there are no universities that devote serious attention to it. What might be more interesting is to prove that it can't be proven using simple mathematics (i.e. prove that only Wiles's proof will suffice. Obviously there will be subtle approaches that may differ, what I mean is proving a general approach ... prove that you must use elliptic curves and modular forms (this would then prove that simple mathematics from fermat's time would not have been sufficient because they didn't have these tools then) )
Good luck
don't waste your time on it though, it emotionally destroyed Wiles, apparently when he was finished it was bitter-sweet (he felt like there was nothing left for him to do)

 

[SteveL27]

Do we know of any collaborative or individual effort to try to prove FLT using only the mathematics known to Fermat at the time? Would any academic institution fund such undertaking or is it considered too whacky?

IH

It's just incredibly unlikely that for 350 years, professional mathematicians missed an elementary solution to the world's most famous unsolved problem. It's far more likely that there is no elementary solution.

Wiles's proof uses highly modern methods that were only developed over the past 50 years or so. To answer the OP's question, to get from "What is a ring?" to understanding Wiles's proof would take at least several years of intensive study, if not more. After all, Wiles was already an established professional mathematician when he started working on FLT, and he then spent seven years on it.

There are a number of popular books on the subject that would be a good place to get an overview of the flavor of the proof. But to grasp the math itself you'd need to know abstract algebra, modern algebraic number theory, and algebraic geometry at the postdoc level. Not anything any of us amateurs are ever going to get close to.

Not meaning to discourage the OP, but the techniques of Wiles's proof are quite advanced and beyond even most professional mathematicians unless they are specialists.

 

[SteveL27]

Don't fool yourself. Remember A.W. spent years of his life solving it, and it practically destroyed him.

You also wrote in another post,

it emotionally destroyed Wiles, apparently when he was finished it was bitter-sweet (he felt like there was nothing left for him to do

I could definitely see someone having a period of creative "post-partum depression" after a monumental achievement like that. I could also see Wiles taking some time off then getting back to seeing if he can find a clearer, simpler proof. Or just kicking back and enjoying life as a genius in Princeton. It's a charming little college town.

Any references appreciated. I did find this, which claims the opposite.

Today Andrew Wiles remains on the faculty at Princeton University. He is involved in research and gives many lectures. He claims that his mind is at rest since he solved Fermat’s last theorem and fulfilled his dream. Wiles will undoubtedly work on and perhaps solve other mathematical problems in his lifetime, but nothing will ever be as special to him as Fermat’s last theorem.

http://mathematics.gulfcoast.edu/mathprojects/andrew_wiles.htm

 

[Bill Simpson]

This book might be appropriate. The first review there sums the book up well.

https://www.amazon.com/dp/0123392519/?tag=pfamazon01-20

The author was the individual responsible for the insight that elliptic curves were key.

But this is not a history of Fermat meant for the general public, this is meant to provide the material for a year of work for a bright undergraduate or graduate student who already has already worked through a year of abstract algebra.