$p$ ramifies in a number field, then it does so in an overfield

Question

If $p$ ramifies in a number field $K$, and we have number field extensions $F:K:\mathbb{Q}$, does it follow that $p$ ramifies in $F$? Please give me some hints. If true, I'll need to work out a direct proof of this.

Answer 1

The ramification index is multiplicative in towers of extensions. If $F/K/Q$ is a tower of extensions, $P_1$ a prime ideal in $Q$, $P_2$ a prime ideal in $K$ with $P_2\cap Q=P_1$, and $P_3$ a prime ideal in $F$ with $P_3\cap K=P_2$, then $e(P_3\mid P_1)=e(P_3\mid P_2)\cdot e(P_2\mid P_1)$.

Answer 2

Let $L/K/\mathbb{Q}$ be an extension of number fields. First notice that if $I$ is an ideal in $O_K$, we can define an ideal in $\mathcal{O}_L$, $I\mathcal{O}_L$ to be the ideal generated by all the elements of $I$. (This notation can, in fact, be used to denote the ideal in $\mathcal{O}_L$ generated by the elements of any set $I$.)

We know we have unique factorisation of ideals into primes, for ideals in $\mathcal{O}_L$ and $\mathcal{O}_K$. Suppose that in $K, (p) =p\mathcal{O}_K = P_1^{e_1}\dots P_n^{e_n}$. Then in $L$, $(p)=p\mathcal{O}_L=(P_1\mathcal{O}_L)^{e_1}\dots (P_n\mathcal{O}_L)^{e_n}$. Whilst the $P_i\mathcal{O}_L$ aren't necessarily primes, the prime factors of $P_i\mathcal{O}_L$ have ramification index at least $e_i$. Thus if $p$ ramifies in $K$, it ramifies in $L$.