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I know about three some good books on the direction: First of all, the book Fermat's Last Theorem by Simon Sin is a pretty good book with the most basic needed materials.
Next, the books 13 lectures on Fermat's last theorem and Fermat's last theorem for amateurs by Ribenboim are pretty well and contain advanced elements.
The last but not the least, the book Fermat's last theorem :a genetic introduction to algebraic number theory is an excellent book by Edwards Harold M which ad hoc adjoins a paper by Kummer, and although it doesn't really solve the problem it provides a well background for it, note that it was published before the theorem was formally proved in 1993.
In general, it is not easy to understand the proof or even to just outline it, while BBC program had produced a video about it to introduce this to the public. Perhaps we can better answer your question provided that you let us know how deeper you want to go in and how much you want to know about the proof exactly. Since you are a junior undergraduate in theoretical physics you must be good in analysis, but what about your algebra? Is it pretty good to go through this?
Anyway, thank you for paying attention.

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I know about three good books on the direction: First of all, the book Fermat's Last Theorem by Simon Sin is a pretty good book with the most basic needed materials.
Next, the books 13 lectures on Fermat's last theorem and Fermat's last theorem for amateurs by Ribenboim are pretty well and contain advanced elements.
The last but not the least, the book Fermat's last theorem :a genetic introduction to algebraic number theory is an excellent book by Edwards Harold M which ad hoc adjoins a paper by Kummer, and although it doesn't really solve the problem it provides a well background for it, note that it was published before the theorem was formally proved in 1993.
In general, it is not easy to understand the proof or even to just outline it, while BBC program had produced a video about it to introduce this to the public. Perhaps we can better answer your question provided that you let us know how deeper you want to go in and how much you want to know about the proof exactly. Since you are a junior undergraduate in theoretical physics you must be good in analysis, but what about your algebra? Is it pretty good to go through this?
Anyway, thank you for paying attention.