TITLE: Weber's class number problem and real quadratic fields of class number one
QUESTION [9 upvotes]: (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?

REPLY [8 votes]: This statement is most certainly nonsense; I guess what she meant to write was that in order to prove the existence of infinitely many number fields with class number $1$ it is sufficient to prove $h(F_n) = 1$ for infinitely many (and therefore for all) $n$.