$L, E\supset K$ are number fields. $L/K$ is normal. And field $M=LE$. Assume $\Omega$ is a prime ideal of M and its intersections with $L, E, K$ are $\mathfrak B,\mathfrak q,\mathfrak p$.
$(1)$ Show that if $\mathfrak B/\mathfrak p$ is unramified then so is $\Omega /\mathfrak q$;
$(2)$ Is that right if $L/K$ is not normal?
$(3)$Show that proposition $(1)$ is still true and consider whether $(2)$ is true if we change all "unramified" into "totally ramified", "inertial" or "splitting completely".
I have studied theory of ramification. So I can understand methods about theory of ramification, Galois theory and commutative algebra. Thanks a lot!