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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?

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I would argue that Riemann-Roch is more of a geometric idea. –  Eugene Jun 12 '12 at 11:52
    
Riemann-Roch is not a number theoretical result but is used to study stuff in number theory. I was drawn to number theory from an early age for the same reasons as you...however to study harder problems we end up constructing more complicated theories. It ultimately takes time but is well worth it...the point of number fields etc is that they all have properties analogous to $\mathbb{Q}$ and contain rings analogous to $\mathbb{Z}$. We use them to study problems about integers by extension to these bigger settings. You should read Stewart/Tall - Algebraic number theory and Fermat's last theorem –  fretty Jun 12 '12 at 13:04
    
'results' such as these are pretty damn nice and definitely algebraic number theory imo. –  Peter Sheldrick Jun 12 '12 at 17:08

5 Answers 5

up vote 10 down vote accepted

Quite a lot of algebraic number theory was invented through trying to prove Fermat's last theorem and other Diophantine problems.

For example, if I asked you to solve the equation $x^2 - y^2 = 5$ in integers it is very simple, you can factorise $(x+y)(x-y) = 5$ and solve the problem by linking to divisors of $5$, in order to get the solutions $x = \pm 3, y=\pm 2$.

Now say I ask you to solve the equation $y^3 = x^2 + 2$ in integers. This is not so easy if we work entirely in $\mathbb{Z}$ and use elementary methods. However, if we shift focus into a bigger ring of numbers, $\mathbb{Z}[\sqrt{-2}] = \{x+y\sqrt{-2}\,|\,x,y\in\mathbb{Z}\}$ then the problem again turns into a multiplicative problem:

$(x + \sqrt{-2})(x- \sqrt{-2}) = y^3$

so that solving the original Diophantine is really the same as solving a "product" style equation in $\mathbb{Z}[\sqrt{-2}]$.

The point of (basic) algebraic number theory is to study rings like this. How do the elements in these rings factorise?

In our problem above it turns out that the ring $\mathbb{Z}[\sqrt{-2}]$ has properties that are strangely close to properties of $\mathbb{Z}$. In fact the elements in this ring "factorise uniquely" into irreducible elements (the analogue of prime numbers in $\mathbb{Z}$).

The phrase "factorise uniquely" does not have quite the meaning you might think, we have to allow for multiplication of units (things that "divide $1$"). It is the ordering of $\mathbb{Z}$ allows us to consider unique factorisations into "positive primes".

There is also a notion of coprimality. This allows us to solve our problem since for odd $x$ it can be shown that $x\pm\sqrt{-2}$ are coprime in $\mathbb{Z}[\sqrt{-2}]$. But their product is a cube so (as in $\mathbb{Z}$), we must have that $x + \sqrt{-2} = (a+b\sqrt{-2})^3$ for some $a+b\sqrt{-2}\in \mathbb{Z}[\sqrt{-2}]$. Comparing coefficients lets you find the possibilities for $a,b$, hence for $x$.

The ideas of Lame and Kummer were to study FLT in the same way by considering the factorisation (for $\zeta$ a primitive $p$-th root of unity):

$z^p = x^p + y^p = (x + y)(x + \zeta y) ... (x + \zeta^{p-1} y)$

forming yet another product equation, now in the ring $\mathbb{Z}[\zeta]$.

Now this is not the entire story, since some of the rings we study in algebraic number theory do not have unique factorisation. For example the ring $\mathbb{Z}[\sqrt{-5}]$ does not since:

$6 = 2\times 3 = (1+\sqrt{-5})(1 - \sqrt{-5})$

gives two totally different factorisations of $6$. Actually the ring $\mathbb{Z}[\zeta]$ does not have unique factorisation for $p=23$, so that FLT could not be solved entirely by the above method.

The thing that stopped factorisation being unique was the fact that the ring wasn't big enough to factorise everything further into the same things.

Fortunately we can restore unique factorisation without having to extend! Through the genius of Kummer and Dedekind, they realised that by considering the "multiples" of an element as an object in its own right, we can reform factorisation in a way that becomes unique upto ordering.

In modern language these objects are called ideals of a ring. There is a notion of a prime ideal, capturing the notion of prime number. The different factorisations of $6$ above can be explained as reordering of the prime ideals in the factorisation of the ideal generated by $6$. These prime ideals are NOT generated by one element, so they dont correspond to "multiples" of something in $\mathbb{Z}[\sqrt{-5}]$, they correspond more to "multiples" of something that doesn't exist in the ring, but would exist after making an extension.

Kummer was able to prove a huge number of cases of FLT by using the ideal theory. This is outlined in many books.

Focus in algebraic number theory now turns to studying these algebraic constructions. We see that in a given "nice" ring, certain prime numbers may factorise, whereas others don't.

For example, in $\mathbb{Z}[i]$ we find that a prime $p$ factorises further if and only if $p=2$ or $p \equiv 1$ mod $4$. The factorisation of $2$ is different to the others in that $2 = (1+i)^2$ is not "square-free". All the others factorise into two different factors. We say that $2$ ramifies, primes $p\equiv 1$ mod $4$ split and primes $p\equiv 3$ mod $4$ are inert.

This congruence relationship describing the factorisation of primes is in some sense really explained by the values of the Legendre symbol $\left(\frac{-1}{p}\right)$, which is also explaining sums of two squares! Working in similar rings gives you the entire quadratic reciprocity law.

The goal of class field theory is to explain the splitting of primes in ANY extension of "number fields" to get similar characterisations in terms of congruences. In fact I just told a lie, we cannot yet do this for ANY extension, class field theory does it for abelian extensions (ones with abelian Galois group) but never-the-less it is quite a strong theory that has many applications (for example it solves the question of which primes can be written as $x^2 + ny^2$.

In the case of abelian extensions of $\mathbb{Q}$ we find that there are simple congruence conditions mod some integer $N$ that completely describe splitting behaviour of primes!

Another side of class field theory is the Cebotarev density theorem, which states essentially that most splitting types happen infinitely often. This is a huge generalisation of Dirichlet's theorem on primes in arithmetic progressions...in fact it provides an infinite amount of Dirichlet theorems, one for each abelian extension.

These days the (mostly unsolved) Langlands program is supposed to be filling in the gaps for non-abelian extensions but this is very difficult to understand and is not yet completely understood. When this is fully understood it will prove to be the holy grail of number theory, it will characterise in a huge way the splitting of primes.

Anyway, I hope this somewhat rushed introduction will whet your appetite. The book I first started with was Stewart/Tall - Algebraic number theory and fermat's last theorem. This is a good book to start you off. Also Lang - Algebraic number theory, Cox - Primes of the form $x^2 + ny^2$ and Childress - Class field theory are good ones to start with for class field theory.

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The oft-quoted assertion that FLT motivated much of the development of algebraic number theory is historically inaccurate - the result of romanticized folklore propagated in popularizations. Rather, much loftier goals served as primary motivation, e.g. pursuit of higher reciprocity laws. For further discussion see here and see here. –  Bill Dubuque Jun 12 '12 at 15:24
    
Oh I know that, but they were still studying FLT were they not? –  fretty Jun 12 '12 at 15:28
    
I just say what I hear/read in books but you have to agree that quite a lot of ANT did arise through the want to solve problems like FLT, whether or not people were directly working on it. –  fretty Jun 12 '12 at 15:32
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Kummer applied the ANT he had developed in connection with reciprocity laws to FLT. Vandiver did not develop ANT. So whom do you have in mind? Certainly not the likes of Dedekind, Hilbert, Weber, Takagi, Artin, Chevalley, or Weil, to name but a few. –  franz lemmermeyer Jun 12 '12 at 18:48
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Well I apologise for being historically inaccurate, I stand corrected. As I say, I go on what I have been told in the past. Mistsakes can be made... –  fretty Jun 13 '12 at 21:52

Here might be one example, although I am not an expert in number theory. If you start to investigate representation of primes by quadratic forms such as $x^2+dy^2$ or $ax^2+bxy+cy^2$, then you can solve the simpler cases using theory which was known back in the days of Euler and Gauss, but to get a deeper picture, and it gets pretty complex (pun intended!), then you really need the power of things like class field theory, complex multiplication, and modular functions. There is a book by D. A. Cox about Primes of the form $x^2+dy^2$ which might give you a good idea. The early parts of it are very readable, and it might provide a nicer introduction than the books you mention.

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I will contribute an example I found here, on page 215.

A positive integer $n$ is a sum of three squares if and only if it is not of the form $4^a(8b−1)$ with $a,b\in \mathbb{Z}$.

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Firstly I'm not sure if you're looking for elementary results that were proven using purely algebraic number theoretic techniques.

One example is Kummer's proof of Fermat's Last Theorem for the case of regular primes, which uses algebraic number theory techniques as opposed to geometric ones.

Let $q \geq 11$ be prime. There are no solutions to $x^q + y^q = z^q$ such that $q \nmid xyz$ and $q \nmid h(\Bbb{Q}(e^{2\pi i/q}))$.

Also there is a discussion related to this on MathOverflow.

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I am looking for such results. What is unclear about the question? I will do my best to edit it. –  Potato Jun 12 '12 at 12:14
    
Does my recent edit make things clear? –  Potato Jun 12 '12 at 12:19
    
Sure. I like good results any way I can get them. –  Potato Jun 12 '12 at 13:04
    
@Potato The mathoverflow link seems to have what you want. –  Eugene Jun 12 '12 at 14:42

I don't know if you're still looking for motivational problems, but I began investigating algebraic number theory through Galois theory and the quest to characterize the algebraic extensions of $\mathbb{Q}$.

I used both of these books to develop mathematical background for the Kronecker-Weber Theorem, which says that any abelian extension of $\mathbb{Q}$ must be cyclotomic. This is a rather beautiful result extending Galois's work, which is a rather easy result from class field theory, although it can also be proved using only ramification theory and valuations (which is why I used Neukirch and Serre as primary references).

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A technical point: the WK thm says that any abelian extension is contained in a cyclotomic extension. Not every subextension of a cyclotomic extension is itself cyclotomic, though. E.g. ${\bf Q}(\sqrt{5})/{\bf Q}$ is a quadratic, cyclic (hence abelian), noncyclotomic extension contained in ${\bf Q}(\zeta_5)$. –  anon May 29 '13 at 19:15

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