Introduction

  • Daniel A. Marcus, "Number Fields", Springer-Verlag.

  • J├╝rgen Neukirch, "Algebraic Number Theory", Springer.

I recommend Marcus' book. Despite the ugly typesetting, the author explains the concepts clearly, and ably motivates the material. Indeed, I’m almost ashamed to admit that until reading the fascinating sections on Fermat’s Last Theorem, abstract algebra was just that to me: abstract. Identifying the bare minimum required for proofs and tweaking rules to see what happens is interesting, but historical background and concrete applications make the subject thrilling.

I now see why mathematicians introduced concepts such as ideals. When attempting to prove results such as Fermat’s Last Theorem, they found themselves working on numbers similar but slightly different to the integers. They succumbed to devious pitfalls when they attempted techniques that worked on integers. They were compelled to question their most basic assumptions.

For example, primes are core features of the integers, abundant in both statements and proofs of countless theorems. But extend the integers a little and abruptly the notions of primes and factorization become fuzzy. To regain focus, mathematicians had to approach from a different angle. Rather than thinking about factors, they’d think about multiples. Rather than think about individual numbers, they would think about sets of numbers at a time (ideals). Instead of working with the number 5, they’d work with the set of numbers that are multiples of 5 (a principal ideal).

They discovered rules for operating on these ideals, making them almost resemble integers. Sometimes, this guided them back to familiar ground. Prime ideals took the place of prime numbers in some proofs. The old trusted methods could be made to work again. And other times, further complications arose, fostering yet more research and discovery.