Let $F$ and $K$ be two number fields with class number one. How can one prove that the class number of $F \cap K$ is also equal to one.
I have been trying to prove something like the intersection of the hilbert class fields is the hilbert class field of the intersection but I haven't been able to do so nor have I any kind of indication that it should be true.
Can we say something in general about the class number of the intersection of two fields ? Or about the Hilbert class field of the intersection of two fields ?
Anyway i'm guessing this assertion must have a way more elementary answer.
The first (non general) question is a consequence of lemma 2.1 of the following notes : https://www.math.ucdavis.edu/~osserman/classes/254a/lectures/35.pdf.
It says that if $K$ is a number field and if $L$ is an extension of $K$ with class number one then the hilbert class field $F\cap K$ is contained in $L$. Then clearly in out case the hilbert class field is contained in both $F$ and $K$ so it is equal to $F\cap K$ which implies that $F \cap K$ has class number one.
I won't accept this answer yet since i'm still looking for a more general one or maybe for a more elementary approach to the first question.