Induced map between class groups injective?

Question

Suppose we have a Galois extension $L/K$ of number fields, and hence an extension $O_L/O_K$ of integer rings. There is a canonical map from the fractional ideals in $O_K$ to $O_L$, just $\mathfrak{p}\rightarrow\mathfrak{p}\cdot O_L$. This obviously takes the principal fractional ideals of $K$ to those of $L$; the generator for $K$ is just the generator for $L$. So it induces a map between ideal class groups $C(K)\rightarrow C(L)$.

My question is: is this map injective? Can we have a non-principal ideal map to a principal ideal, pulling some generator out of nowhere? I know that in the case of orders, it's not true: for example, we look at $\mathbb{Z}[\sqrt{-3}]\rightarrow\mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$ and the map of ideals $(2, 1+\sqrt{-3})\rightarrow(2)$. In this case $(2, 1+\sqrt{-3})$ is not principal. Is there some property of Dedekind domains that makes this true? Or is there some counterexample? I just don't know too many class groups, and all the ones I know are trivial quadratic extensions, so it is useless in verifying this conjecture.

Answer 1

The map need not be injective. In fact, if $L$ is the Hilbert class field of $K$, then every ideal of $K$ will be principal in $L$. The Wikipedia page on Hilbert class fields has more information, as well as an explicit example.

Answer 2

Noting $G=Gal(L/K)$, your induced map $C(K)\to C(L)$ actually takes its values in the $G$-invariant subgroup $C(L)^G$. Usually called "transfer" (this comes from the cohomological formulation of CFT) or "capitulation" map, denoted $cap_{L/K}$, it is in general neither injective nor surjective. There is a considerable literature devoted to the study of the kernel and cokernel of $cap_{L/K}$, within the so called "genus theory" of class groups. A typical result is Chevalley's formula for "ambiguous classes" (1933) when $L/K$ is cyclic, which expresses the quotient of the orders of $C(L)^G$ and $C(K)$ in terms of indices of norm groups inside groups of units. But things get much more complicated in the general case, where $cap_{L/K}$ takes place in longer and longer exact sequences (up to 7 terms). I don't know how to attach a file on SE, but if you are interested, just tell me how, and I'll try to append a synthetic article on the "state of the art" concerning this subject.