Question

A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$.

What can be said about the parity (odd or even) of the class number of a pure cubic field? In particular, is there some theorem (or simple algorithm) that tells you when the class number will be odd?

I have found this paper http://www.rzuser.uni-heidelberg.de/~hb3/publ/pcub2.pdf of Lemmermeyer, but it doesn't seem to have what I'm looking for.

I'm also interested in finding some good references with information relating to my questions or on the theory of cubic fields.

Answer 1

There is an algorithm for computing the 2-class number of pure cubic fields using $2$-Selmer groups of elliptic curves due to G. Frey and his diploma students Eisenbeis and Ommerborn: Computation of the 2-rank of pure cubic fields, Math. Comput. 32 (1978), 559-569.

This was generalized by U. Schneiders in Estimating the 2-rank of cubic fields by Selmer groups of elliptic curves, J. Number Theory 62 (1997), No.2, 375-396; the proof of the upper bound on the 2-class number in this article is, however, incorrect.

This being said, the 2-class number of pure cubics is as mysterious as the 3-class number of quadratic number fields. In particular, there is no genus theory for the 2-part of the class number.