(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](http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/Yukako_Kezuka_MSc_Project.pdf), 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?



Weber's class number problem and real quadratic fields of class number one
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) $n$. 


