The purpose of this question is to motivate me to study algebraic number theory.
Let me explain. My motivation for studying number theory is to learn about beautiful results with simple, accessible statements. For example, the theorem that a prime can be written as the sum of two squares if and only if it is 1 mod 4. I have been reading the first few pages of both Neukirch's Algebraic Number Theory and Serre's Local Fields. I have looked ahead to see what I have to look forward to in terms of such results.
- A prime is the sum of two squares if and only if it is 1 mod 4 (done in the first few pages by describing characterizing factorization in the Gaussian integers)
- Pythagorean triples (an exercise in the first section)
- Solving Pell's Equation
- Quadratic Reciprocity (using a result about cyclotomic number fields)
- Lagrange's four-square theorem
All of these appear relatively early in the book, and are provable with elementary number theory accessible to a motivated high school student. Looking at some of the later chapters, I wonder: What is the point of all this theory, besides being aesthetically pleasing?
I do not wish to denigrate the absolutely beautiful theory contained in these books. It is marvelous, and I want to learn it. However, I wonder how it can be applied to produce results on the standard integers, especially ones not provable by elementary methods. Fermat's Last Theorem is the obvious example, but surely there are others. A reference where I could find a collection of such results would be especially appreciated. In case it is not clear, I am looking for results provable with methods and theorems developed in the books I mentioned above (or other similar standard graduate texts).
Perhaps the problem is that I don't yet appreciate that results about algebraic integers and number fields can be as thrilling as those about integers. If this is the case, I'm interested in seeing a short list of remarkable theorems about such objects. Maybe I just need to refine my tastes.
To summarize: How can the powerful theorems of algebraic number theory (e.g. the version of Rieman-Roch found in Neukirch, Class Field Theory) be used to give interesting elementary results?
