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November 23rd, 2011, 08:40 AM   #1
 
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FLT book

Does anyone know a good book on the mathematics that Andrew Wiles used to solve Fermat's Last Theorem?
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November 23rd, 2011, 11:35 AM   #2
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Re: FLT book

At what level? Going from a high school diploma to understanding the Wiles-Taylor paper could take 15 years of study... but if you just wanted to know the basics of modular curves that would be a simpler task.

(Of course if you want something fairly in-depth I won't be much help since I'm not an expert in the area.)
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November 23rd, 2011, 11:54 AM   #3
 
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Re: FLT book

I study mathematics at my university, I already have some basic knowledge of mathematics and in some fields more advanced. I just want to know what branches of mathematics he used for solving Fermat's Last Theorem not only the basics. And then I will try to understand them, I want to know what are the "foundations" first.
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November 23rd, 2011, 01:39 PM   #4
 
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Re: FLT book

So, as CRG intimated, both your end goal and your current level of mathematics are very important in making any recommendations. If your goal is a full understanding of the proof of Fermat's Last Theorem, a word of caution for your expectations: there are probably fewer than 100 people in the world who fully understand all of the details in the proof (I suspect 100 is a gross overestimate).

The notion of "foundations" is rather vague and dependent on just how much you want to know. The full "foundations" for a complete understanding of the proof include a PhD in arithmetic geometry.

What kind of mathematics do you know? Certainly any attempt to understand even the big picture of the FLT proof requires a solid understanding of graduate abstract algebra, at least some sort of graduate number theory, and comfort with complex analysis.

I will direct you attention to an older thread:
http://www.mymathforum.com/viewtopic.php?f=40&t=22786

In particular, I would recommend reading the posts by Peter for insight on where to focus your attention once you feel like your general mathematics is at the graduate level.

For a basic background, the Wikipedia articles are worth reading to get an sense of the basic vocabulary and ideas (and are free):
http://en.wikipedia.org/wiki/Wiles%2...s_Last_Theorem
http://en.wikipedia.org/wiki/Fermat'..._general_proof

Ultimately, I would discuss this question with a number theory professor or post-doc at your university, as they will have a better idea of what you know and how that stacks up against what you need to know to get where you want to get. Perhaps even more importantly, they can gauge your actual interest. With any but the most modest of goals, if you are not passionate about doing this, you will not succeed.
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November 23rd, 2011, 02:30 PM   #5
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Re: FLT book

Quote:
Originally Posted by Turgul
So, as CRG intimated, both your end goal and your current level of mathematics are very important in making any recommendations. If your goal is a full understanding of the proof of Fermat's Last Theorem, a word of caution for your expectations: there are probably fewer than 100 people in the world who fully understand all of the details in the proof (I suspect 100 is a gross overestimate).
I wouldn't be surprised if it was as many as a hundred now. There has been a lot of work done to make the work accessible and to simplify complex parts. But I wouldn't guess much higher!

But it's entirely possible that this is not what the OP wants. Perhaps just an outline of algebraic geometry and abstract algebra, plus a 'path' of how the pieces add up to the full theorem.
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November 24th, 2011, 07:31 AM   #6
 
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Re: FLT book

Quote:
Originally Posted by Turgul
So, as CRG intimated, both your end goal and your current level of mathematics are very important in making any recommendations. If your goal is a full understanding of the proof of Fermat's Last Theorem, a word of caution for your expectations: there are probably fewer than 100 people in the world who fully understand all of the details in the proof (I suspect 100 is a gross overestimate).

The notion of "foundations" is rather vague and dependent on just how much you want to know. The full "foundations" for a complete understanding of the proof include a PhD in arithmetic geometry.

What kind of mathematics do you know? Certainly any attempt to understand even the big picture of the FLT proof requires a solid understanding of graduate abstract algebra, at least some sort of graduate number theory, and comfort with complex analysis.

I will direct you attention to an older thread:
http://www.mymathforum.com/viewtopic.php?f=40&t=22786

In particular, I would recommend reading the posts by Peter for insight on where to focus your attention once you feel like your general mathematics is at the graduate level.

For a basic background, the Wikipedia articles are worth reading to get an sense of the basic vocabulary and ideas (and are free):
http://en.wikipedia.org/wiki/Wiles%2...s_Last_Theorem
http://en.wikipedia.org/wiki/Fermat'..._general_proof

Ultimately, I would discuss this question with a number theory professor or post-doc at your university, as they will have a better idea of what you know and how that stacks up against what you need to know to get where you want to get. Perhaps even more importantly, they can gauge your actual interest. With any but the most modest of goals, if you are not passionate about doing this, you will not succeed.
I know calculus, topology, basic algebra and Galois theory, Analytic Geometry, Probability, statistics, differential geometry and set theory, and i know some basic on category theory and Homology algebra.
I have asked my teacher at university who is interested in number theory but he didn't know much.
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November 25th, 2011, 11:41 PM   #7
 
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Re: FLT book

It is hard to tell exactly what you are looking for, but these might be in the right direction:

Mozzochi's The Fermat Proof
Hellegouarch's Invitation to the Mathematics of Fermat-Wiles

Mozzochi's book is very brief (~30 pages of exposition) and aims to give an idea to the basic mathematics involved in the proof. There are no details here, but if you are only looking for a general idea of what is involved this may be enough. It should certainly be approachable at your level, especially with the help of Wikipedia for elaboration on any of the new mathematical ideas presented. It also sports a very large bibliography (though beware, most of the listed items are quite technical). I imagine you could tackle this book in a day or two. If your curiosity about the proof is just that, a curiosity, this may be exactly what you are looking for.

Hellegouarch's book is much more substantial (~350 pages) and aims to get you acquainted, in a non-trivial way, with the mathematical framework of the proof. As far as I can tell, it is the most elementary of the books with this aim. The beginning of this book should be approachable, and there is a great deal to learn by working through it. Where Mozzochi's book can be managed in a day or two, this book will take months of study to get through. Without additional coursework in the plethora of related fields, you may not be able to finish this book on your first pass. But it will help you get a much deeper understanding of the mathematics involved and may help motivate the math you will learn in other classes. This is probably the place to start if your goal of understanding the FLT proof is serious.

Should you make it through Hellegouarch, of course, the rabbit hole goes deeper.
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November 26th, 2011, 12:09 AM   #8
 
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Re: FLT book

ok, thanks a lot. I will try this books at first.
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November 26th, 2011, 12:35 PM   #9
 
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Re: FLT book

Hopefully they help get you in the right direction. If you are interested in number theory in general, I also recommend Ireland and Rosen's A Classical Introduction to Modern Number Theory as my favorite introduction to many of the ideas which you will have to learn sooner or later if you want to do serious study into number theory. It may prove a helpful supplement to Hellegouarch's book.
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