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.

yourundergraduate course? – KConrad May 17 '10 at 8:51