# Parity of class number of pure cubic fields

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.

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This probably doesn't help you much, but The 2-part of the pure cubic field is equal to the 2-part of its Galois closure. – Alex B. Feb 5 '12 at 9:01

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.