Math 254A, UC Berkeley, Fall 2001 (Kedlaya) Lecture 10: Fun Facts about Class Numbers September 19, 2001

Reference. Neukirch, Section 1.6.

Outline of lecture.

1. General warning: if K ⊂ L are number field, there is a map Cl(K) → Cl(L) induced by the map of ideal groups sending an integral ideal 𝔞 to 𝔞𝔬_{L}. But the map on class group is not injective! E.g., ℚ(\sqrt{−5}) ⊂ ℚ(\sqrt{−1}, \sqrt{−5}), the former has class number 2, the latter has class number 1. See also the homework.

2. The class number of the imaginary quadratic field ℚ(\sqrt{−D}) is bounded above by O(\sqrt{D}), and is bounded below by O(D^{1/2−ϵ})—but the lower bound is not effective. (Siegel) The best effective known lower bound is O((log D)^{1−ϵ}) (Goldfeld).

3. The only imaginary quadratic fields ℚ(\sqrt{−D}) of class number 1 are for D = −1, −2, −3, −7, −11, −19, −43, −67, −163. The complete list of class number N is known for some other small values of N, using Goldfeld’s lower bound.

4. On the other hand, it is conjectured that there are infinitely many real quadratic fields ℚ(\sqrt{D}) of class number 1. The upper bound O(\sqrt{D}) is thus much further from the truth in this case.

5. There are conjectures (Cohen-Lenstra) about the probability that the class number of a quadratic field (real or imaginary) is divisible by a prime p. For p = 2, this is understood by work of Gauss (see homework). For p = 3, this is basically understood by Davenport-Heilbronn. Beyond that, we know almost nothing.

6. For cyclotomic fields ℚ(ζ_{n}), the class number grows with n, but is much harder to compute than in the cyclotomic case. However...

7. The prime p is called regular if p does not divide the class number of ℚ(ζ_{p}). A theorem of Kummer states that p is regular if and only if p does not divide any of the numerators of the Bernoulli numbers B_{2}, B_{4}, …, B_{p−3}. (A more precise result is due to Herbrand and Ribet.)

8. Kummer proved Fermat’s Last Theorem for regular primes. The basic idea is that if y^{p} = z^{p} −x^{p} = (z −x)(z −ζx) ⋯ (z −ζ^{p−1}x), then each of the ideal classes (z −x), (z − ζx), …, (z −ζ^{p−1}x) is a p-th power in the class group. But if p is regular, then an ideal whose p-th power is principal is itself principal!

9. It is conjectured (Vandiver’s conjecture) that the class number of the maximal real subfield of ℚ(ζ_{p}) is never divisible by p.