What fields of number theory do I need to learn to understand Wiles' proof?

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In summary, to fully understand the proof of his and Taylor's theorem, it would be helpful to start with the book "Fermat's Last Theorem" by Springer, which includes a comprehensive bibliography for further learning. Some recommended authors to consider are Silverman and the standard mathematical pedagogy may not always be the most efficient way to learn specific concepts. It may be more helpful to focus on learning specific topics, such as basic graduate level algebraic and analytic number theory, rather than trying to cover a broad range of essentials.
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MathematicalPhysicist
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What fields of interest in number theory should I learn to fully understand his and Taylor's proof?
Is there a comprehensible list?
 
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  • #2
Start with the book of the proof (Springer, I think). It will have a comprehensive bibliography for when you need to recursively learn something.
 
  • #4
Read the reviews, decide which one's best for you. Silverman is a good author, for what it's worth.
 
  • #5
matt grime said:
recursively learn something.

Kinda goes against the standard math pedagogical philosophy. I bet when machines go sentient that's how they'll learn.
 
  • #6
The mathematical pedagogy to which you refer is aimed at teaching a broad canvas of essentials. This is good if someone has decided what those are on your behalf.

Sometimes, though, if you need to just learn something, say to understand a paper or a particular result, you need to find out what you need to learn, and this is perhaps a case in point.

To learn basic graduate level algebraic and analytic number theory would be a few years, from scratch. And then you find out that you spent too long on cubic reciprocity when you needed to learn more about L functions and modular forms..
 

1. What is Andrew Wiles' proof?

Andrew Wiles' proof, also known as the Wiles-Taylor conjecture, is a mathematical proof that solves Fermat's Last Theorem, a famous problem in number theory that remained unsolved for over 300 years.

2. How did Andrew Wiles prove his theorem?

Wiles used a technique called "modular forms" to prove his theorem. He combined this with other mathematical concepts such as elliptic curves and Galois representations to develop a proof that is now considered one of the most brilliant achievements in mathematics.

3. How long did it take Andrew Wiles to prove Fermat's Last Theorem?

Wiles spent over 7 years working on the proof, starting in 1986 and finally announcing his solution in 1993. However, it was later found that there was a gap in the proof, and it took Wiles and his colleague Richard Taylor another year to fill in this gap and complete the proof.

4. What impact did Andrew Wiles' proof have on mathematics?

Wiles' proof of Fermat's Last Theorem had a huge impact on mathematics. It opened up new areas of research and sparked interest in number theory and its applications. It also solidified Wiles' reputation as one of the greatest mathematicians of our time.

5. Is Andrew Wiles' proof considered to be a major achievement in mathematics?

Yes, Wiles' proof is considered to be one of the most significant achievements in mathematics. It solved a problem that had baffled mathematicians for centuries and has been hailed as a masterpiece in the field. It also earned Wiles numerous prestigious awards and accolades.

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