How important is getting nitty-gritty with ideals for algebraic number theory?

Question

Coming off an undergraduate course on number fields based on Marcus's textbook Number Fields, I am interested in taking the logical next step towards (local) class field theory, as well as Iwasawa theory. I had some difficulty with working with ideals and their prime decomposition in number rings for the first time: e.g., trouble with ideal operations, not really understanding how to compute the finite field $\mathcal{O}/\mathfrak{p}$, the phenomenon of inertia as it relates to $[\mathcal{O}_L/\mathfrak{q}:\mathcal{O}_K/\mathfrak{p}]$.

I would like to eventually get better at working hands-on with ideals, e.g., computing the ideal class group of a number field using the Minkowski bound. However, I'm not sure how much intuition I should have for ideals in number rings (and Dedekind domains, more generally), as I go into algebraic number theory 'proper', i.e., CFT. Should I ground my intuitions in more classical settings before wading through the abstractions (idelic CFT, Galois cohomology, etc.)? Or would I be better served by picking it up along the way?

Answer 1

You don't need to be able to compute an ideal class group by hand or know how to put ideal classes to work in solving a problem (like determining the integral solutions to some Mordell equation) in order to learn algebraic number theory, but then it is likely you won't grok what ideal classes really are or why they are worthwhile. Would you want to learn the residue theorem in complex analysis just as some abstract thing, without ever seeing how it lets you compute real integrals? Or learn algebraic topology without being able to compute some homology or cohomology groups to tell some spaces apart?

Several quotes come to mind when reading your post.

It seems that over the years, everything that has been done has proved useful, theoretically or as examples, for the further development of the theory. Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more.

Serge Lang, Foreword to Algebraic Number Theory (1st ed.)

It is a depressing fact that many number theorists have never acquired sufficient technique to perform number theoretic calculations in anything but a quadratic field.

A. Fröhlich and M. J. Taylor, Preface to Algebraic Number Theory (1st ed., 1991)

When general theory proves the existence of some construction, then doing it in terms of explicit coordinate expressions is a useful exercise that helps one to keep a grip on reality, [but] this should not however be allowed to obscure the fact that the theory is really designed to handle the complicated cases, when explicit computations will often not tell us anything.

Miles Reid, p. 117 in Undergraduate Algebraic Geometry

Although it is usually simpler to prove a general fact than to prove numerous special cases of it, for a student the content of a mathematical theory is never larger than the set of examples that are thoroughly understood.

Vladimir Arnold, p. vi in Lectures on Partial Differential Equations

A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one.

Paul Halmos, p. 63 in I Want to be a Mathematician: An Automathography

When Fröhlich and Taylor wrote their algebraic number theory book, computational number theory packages were much less widely available (and powerful) than they are today. An updated version of their quote above might stress the utility of being able to do computations with a computer algebra system, since numerical data are important as a way of testing ideas.

Answer 2

As always, having been exposed to a number of examples and concrete situations helps understanding what is going on in more general and abstract settings, so be it with Algebraic Number Theory (ANT).

Yet I am not sure that working out a huge amount of, say, explicit computations of ideal class groups really helps understanding what is going at a higher level. For instance, when working out the details of an ideal class group the problem of deciding about the existence of an element of given norm gets intricated when the degree of the extension is $\geq3$ and in itself has no relevance in Class Field Theory or other higher topics.

Generally speaking my impression is that in ANT being able to compute explicit examples requires techniques and mathematical tools that are not necessary to understand the general theory. Another example is given by the problem of finding a basis for a lattice in $\mathbb{R}^n$ which is not exactly a trivial task.

I would venture to say that in order to tackle the general theory a good undestanding of the situation for quadratic extensions and the cyclotomic fields $\mathbb{Q}(\zeta_n)$ is enough. Of course there are available in the literature many more other cases and one can take a look at those too as study progresses just to have a more concrete grip at things.

I would pay more attention at trying to draw parallels with geometry and understanding how classical "elementary" results are instances of more general theorems, e.g. Gauss' reciprocity law being a special case of Artin's reciprocity law.