[Math] care about fields of positive characteristic

abstract-algebrafield-theory

This is what I know about why someone might care about fields of positive characteristic:

  1. they are useful for number theory
  2. in algebraic geometry, a theory of "geometry" can be developed over them, and it's fun to see how this geometry works out

Some people might read this and think, "What more could you need?" But I've never been able to make myself care about number theory, so (1) doesn't help me. (2) is nice for what it is, but I'm hoping there's something more. My understanding of (2) is that this is only geometry in a rather abstract sense and, for instance, there's no generally useful way to directly visually represent these fields or varieties over them the way we can over the reals or complex numbers. (Drawing a curve in R^2 and saying it's the curve over some other field may be helpful for some purposes, but it's not what I'm after here.)

Is there anything else? The ideal (surely impossible) answer for me would be "Yes, such fields are very good models for these common and easy to understand physical systems: A, B, C. Also, we can visualize them and varieties over them quite easily by method D. Finally, here's a bunch of surprising and helpful applications to 500 other areas of mathematics."

UPDATE: to answer Qiaochu's comment about what I do care about.

Let's say I care about:

algebraic & geometric topology

differential geometry & topology

applications to physics

and I certainly care about algebraic geometry over C

(this is to say I understand the motivations behind these subjects and the general idea, not necessarily that I know them in depth)

Best Answer

Let's suppose you care about finite groups, one way or another. Then you probably care about the classification of finite simple groups. The bulk of this classification is the groups of Lie type, which were discovered by finding analogues of Lie groups over finite fields.

Finite fields are also (as one might guess) very important in computer science. I'm certainly no expert, but here are some applications I know of:

  • Cryptographic protocols like Diffie-Hellman have as their basis the simple fact that it is difficult to invert exponentiation in finite fields.
  • The standard way that one factors polynomials over the integers is to apply something like Berlekamp's algorithm to factor them over several finite fields first, then combine the factorizations.
  • The classic theorem that IP=PSPACE requires some work over finite fields.
  • Elliptic curves over finite fields are used for elliptic curve cryptography.
  • I have also been told that vector spaces and varieties over finite fields can be used to construct error-correcting codes. I don't know anything about this, but here is a book on the subject. For the special case of linear codes this leads to a beautiful analogy between lattice sphere packings and error-correcting codes which is described, for example, by Noam Elkies here.

Finally, even if you are only interested in varieties over $\mathbb{C}$ (say), if your variety happens to also be nice and defined over $\mathbb{Z}$ then it can be nice and defined over $\mathbb{F}_p$ for all but finitely many $p$ and you can use the Weil conjectures to compute its Betti numbers by counting. This is particularly easy to do for varieties with nice moduli interpretations like flag varieties.

Edit: You might also be interested in reading Serre's expository article How to use finite fields for problems concerning infinite fields, as well as Manin's Reflections on arithmetical physics. I got the latter link from an excellent answer to an MO question on mirror symmetry over finite fields.

Related Question