TITLE: Transitivity of the discriminant of number fields
QUESTION [6 upvotes]: Let $M/L/K$ be a tower of number fields with discriminant of $M/K: d_M$ and of $L/K: d_L$. I would like to find a transitivity theorem for the discriminant and by letting $p_i$ and $q_i$ be integral basis for $M$ and $L$ respectively and $A =[a_{ij}]$ the transition matrix between the basis, a calculation gives:
$$[M:L]^{[L:K]}d_L = \det(A)^2d_M$$
However, these two links give different(even from each other) answers:
Divisibility of discriminants in number field extensions $([M:L]^2 d_L = \det(A)^2 d_M)$
Quadratic subfield of cyclotomic field (discriminant of $M$ is divisible by discriminant of $L$ to the power $[M:L]$
Both of these are given in the accepted answers and use different notation. Which of the three is correct?(The last one is not strictly contradictory but probably often will be...).

REPLY [7 votes]: A couple of definitions are in order to clarify the issue.
Definition 1. Consider a field extension $L / K$ and a basis $\{\alpha_1,\dotsc,\alpha_n\} \subset \mathcal{O}_L$ of $L$ over $K$. Its discriminant is
$$
d(\alpha_1,\dotsc,\alpha_n) = \det(\text{Tr}\,(\alpha_i\alpha_j))
$$
Definition 2. The discriminant $\mathfrak{d}_{L/K}$ of a field extension $L/K$ is the ideal of $\mathcal{O}_K$ generated by the discriminants $d(\alpha_1,\dotsc,\alpha_n)$, where $\{\alpha_1,\dotsc,\alpha_n\}$ ranges over all the bases of $L/K$ with elements in $\mathcal{O}_L$.
Note that if $\alpha_1,\dotsc,\alpha_n$ generate $\mathcal{O}_L$ over $\mathcal{O}_K$ as a free module, then $\mathfrak{d}_{L/K} = d(\alpha_1,\dotsc,\alpha_n)\mathcal{O}_K$, because then all other bases can be obtained transforming $\alpha_1,\dotsc,\alpha_n$ by an appropriate matrix with entries in $\mathcal{O}_K$.
Clearly $\mathfrak{d}_{L/K}$ is always principal if $\mathcal{O}_K$ is a PID, e.g. if $K = \mathbb{Q}$, but it may not be in general.
Finally, we have the following:

Theorem. If $K \subseteq L \subseteq M$ is a tower of fields, then
  $$
\mathfrak{d}_{M/K} = \mathfrak{d}_{L/K}^{[M:L]} N_{L/K}(\mathfrak{d}_{M/L})
$$
  Proof. See Neukirch's Algebraic Number Theory, corollary 2.10, chapter III.