prime ideal of $\mathcal O_K$ splits completely in tower of Galois extensions The following exercise is taken from D.A.Cox's book "Primes of the form $x^2 +ny^2...$"
He uses this in order to prove some intermediate steps before the proof of the main theorem in chapter 5, which solves the problem of determing which primes can be expressed in the form $x^2 + ny^2$, for infinitely many n's (theorem 5.1).

If $ K \subset M \subset L $ , where $L$ and $M$ are Galois over $K$,
  then prove that a prime ideal $ \mathfrak p $ of  $ \mathcal O_K$
  splits completely in $L$ if and only if it splits completely in $M$
  and some prime of $\mathcal O_M $ containing $\mathfrak p $ splits
  completely in $L$.

Any help would be really appreciated.
Thank you in advance.
 A: The key is the multiplicativity of the residue class degree and ramification index in towers.  I don't see where Cox covers it (n.m. it's in the exercises), but it's in any other ANT book.  You have $e_{\mathfrak Q | \mathfrak p} = e_{\mathfrak Q|\mathfrak P} e_{\mathfrak P|\mathfrak p}$ and $f_{\mathfrak Q | \mathfrak p} = f_{\mathfrak Q|\mathfrak P} f_{\mathfrak P|\mathfrak p}$ when $\mathfrak Q \supset \mathfrak P \supset \mathfrak p$ is a chain of primes lying in $\mathcal O_L \supset \mathcal O_M \supset \mathcal O_K$ respectively.  To split completely these numbers all need to be $= 1$, and by these formulae $f_{\mathfrak Q | \mathfrak p}$ is $= 1$ iff $f_{\mathfrak Q|\mathfrak P}$ and $f_{\mathfrak P|\mathfrak p}$ are both $= 1$, and the same for $e$.
In a nutshell, residue degree is multiplicative because $[\mathcal O_L /\mathfrak Q: \mathcal O_K / \mathfrak p] = [\mathcal O_L /\mathfrak Q: \mathcal O_M / \mathfrak P][\mathcal O_M /\mathfrak P: \mathcal O_K / \mathfrak p]$ and ramification is multiplicative by exercise 5.15, which you must have done since you're on 5.18 :p
