John Carlos Baez on Nostr: Mathematicians number our theorems because... well, we can buy numbers for cheap from ...

Mathematicians number our theorems because... well, we can buy numbers for cheap from the wholesaler, so we have a lot of them lying around, and we use them for everything. But it's very rare for the number of a theorem to become famous. The only exception that leaps to my mind is Hilbert's Theorem 90. Do you know any others?

If you ask a number theorist about Hilbert's Theorem 87, they'll look at you like you're nuts. But if you ask about Theorem 90 they'll know it, or at least be embarrassed for forgetting the exact statement. Try it!

It was the the 90th theorem in Hilbert's paper "Theory of algebraic number fields", and for some reason it became more famous than most. So what does it say?

Here's an easy special case. The real numbers ℝ sit inside the complex numbers ℂ, and the complex numbers have exactly two symmetries that leave every real number unchanged. Complex conjugation is the interesting one, and the other is the identity. If we take a complex number z, apply each of these symmetries, and multiply the results, we get

zz*

Hilbert's Theorem 90 says that if zz* = 1, then z = a*/a for some nonzero complex number a.

But his theorem is a lot more general! It applies whenever you've got a field F sitting in a larger field K, and the symmetries of K that leave all elements of F unchanged form a finite group generated by some symmetry σ.

Here's what it says. Take an element z of K, apply all these symmetries to it, and multiply the results. If you get 1, then we must have z = σ(a)/a for some nonzero a in F.

But what's really going on here???

(1/2)