Andrew Wiles proof.

[MathematicalPhysicist]

What fields of interest in number theory should I learn to fully understand his and Taylor's proof?
Is there a comprehensible list?

[matt grime]

Start with the book of the proof (Springer, I think). It will have a comprehensive bibliography for when you need to recursively learn something.

[MathematicalPhysicist]

http://www.amazon.com/s/ref=nb_ss_b?url=search-alias%3Dstripbooks&field-keywords=fermat%27s+last+theorem+springer

Which one of them are you reffering to?

[matt grime]

Read the reviews, decide which one's best for you. Silverman is a good author, for what it's worth.

[Dragonfall]

recursively learn something.

Kinda goes against the standard math pedagogical philosophy. I bet when machines go sentient that's how they'll learn.

[matt grime]

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..