(A) It is an old, outstanding problem to show that there are infinitely many real quadratic fields with class number one.

(B) On the other hand, Weber's class number problem (for $p=2$) asks to show that the degree $2^n$ cyclotomic extension $F_n=\mathbb Q(\cos(2\pi/2^{n+2}))$ of $\mathbb Q$ has class number one, for all $n$.

According to p.41 of this thesis, showing (B) for infinitely many $n$ implies (A), but no idea of the proof is given. This does not seem to follow immediately for me, but perhaps I am missing something simple here. Certainly $F_n$ contains a real quadratic subfield, but why should its class number also be 1?

too broadas a hypothesis. For example, if GRH is false for the zeta-function of $\mathbf Q(\sqrt[3]{2})$, Heilbronn can't deduce anything about class numbers of imaginary quadratic fields. The correct hypothesis is failure of a restricted version of GRH: just for $L$-functions of imaginary quadratic (Dirichlet) characters. $\endgroup$