I'm interested in learning about any applications, the more worldly the better*.
Pointing to a nice reference on the number field sieve, for example, would be fine.
However, let me mention one direction I would be especially grateful to learn about.
In my introductory course, I like to spend some time on the perspective that algebraic number theory
is the study of sophisticated multiplications on $\mathbb{Q}^n$ (an algebraic number field $F$ of
degree $n$) and on $\mathbb{Z}^n$ (the ring of algebraic integers in $F$).
This is in part because I still find it amazing that a little bit of abstract algebra (of irreducible polynomials)
enables us to construct such things systematically**.
But I also believe at least half-seriously that this is the view through which the general public will
gradually learn about algebraic numbers, until the time they're taught in primary schools several thousand years hence.
After all, we have ourselves witnessed the remarkable ascent of multiplication on $\mathbb{F}_2^n$, a set whose initial practical use was entirely devoid
of algebraic content, as a powerful tool
for information processing.
After such grandiose reflection, I can't help but wonder: are multiplicative structures on $n$-tuples of integers provided
by algebraic number theory already of some practical use? A superficial google search uncovered nothing.
But surely, there must be something? I would love to be able to mention some examples to my students.
As I write, one class of examples occurs to me. Algebraic integers can be used to construct arithmetic groups, which I understand can be applied in a number of ways. Perhaps someone can comment knowledgeably on that. But something direct that could
at least vaguely be explained in an undergraduate course would be even better.
Added:
Such was the depth of my ignorance that I didn't even know about number field codes until Victor Protsak pointed to them in his answer. Thanks to him, I stumbled upon a short survey by Lenstra. To get the gist of it, one need only read this quote:
'The new codes are the analogues, for number fields, of the codes constructed
by Goppa and Tsfasman from curves over finite fields.'
The time-worn analogy continues to prosper.
Added again:
In order to avoid misleading people with the word 'prosper,' I should say that Lenstra has many negative things to say about these codes. For example,
'If the generalized
Riemann hypothesis is true our codes are, asymptotically speaking, not as
good as those of Goppa and Tsfasman Also, the latter codes are linear and
non-mixed.'
My original question still stands.
*I do not wish, however, to give the impression of a firm belief in the division
between pure and applied mathematics.
** To appreciate this, one need only spend a little time
on a direct approach using the multi-linear algebra of structure constants.
Best Answer
Your requirements are quite stringent! As you know well, ANT is a couple of layers removed from "practice". In general, I find that the methods deriving from the development of algebraic number theory eventually lead to incomparably more applications than any of the standard ANT theorems themselves. Just a few examples that quickly spring to mind: Gauss reduction of quadratic forms → shortest lattice vectors → LLL; Dirichlet units → Minkowski geometry of numbers → convex geometry; class groups and unit groups → finitely generated abelian groups → (pick your favorite application of group theory, e.g. in abelian harmonic analysis). I would try to project this deeper idea over the immediate payoff. Also, unsurprisingly, "elementary" number theory presents more immediate applications, e.g. to cryptography and specifically, to primality testing and factorization.
But enough of philosophy! Here a few concrete applications:
Construction of codes and dense lattice packings using multiplicative groups of global fields by Rosenbloom and Tsfasman (Invent. Math. paper or see Tsfasman's survey "Global fields, codes and sphere packings").
Margulis arithmeticity theorem: not only are algebraic integers useful in constructing discrete groups, but after imposing certain conditions (an irreducible lattice in a higher rank semisimple Lie group), all of them arise in this way! These lattices have been used in constructing Ramanujan graphs and superconcentrators and to uniform distribution of points on spheres (admittedly, arithmetic of the field is rather secondary: you can get good constructions starting from a group over Z).
Lind's theorem on realizability of any Perron–Frobenius integer as the leading eigenvalues of a positive integer matrix. Since log λ is the entropy, this does have semi-practical consequences (cf the textbook of Lind and Marcus).
Lucas–Lehmer primality test, Lucas and Fibonacci pseudoprimes, Grantham's Frobenius test. This strides the border between elementary and algebraic number theory, which may actually be an advantage in an undergraduate class!
I'd be curious to know how do they jibe with your goals.