Given two number fields $E$ and $F$, is there a bound on $|d_{EF}|$, the absolute value of the absolute discriminant of the compositum of fields $EF$, in terms of $d_E$, $d_F$, and the extension degrees ?
The actual problem I try to solve is slightly more specific: given a number field $F$, is there a bound on $|d_{F^n}|$ in terms of $d_F$ and the extension degrees, where $F^n$ is the normal closure of $F$ over $\mathbb Q$? We know that $F^n$ is the compositum of all the conjugates of $F$ over $\mathbb Q$, and in the particular case I study I know that there are at most $4$ of those conjugates. So a bound on the discriminant of any compositum of fields would help, but if there is a bound for the discriminant of the normal closure it would be even better.
