the ramification index and the inertial degree are multiplicative in towersRamificationIndexAndTheInertialDegreeAreMultiplicativeInTowers2005-03-03 17:19:452005-03-03 17:22:04Theoremsplitting and ramification in number fields and Galois extensionstowers of number fieldsramificationinertia% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\intF}{\mathcal{O}_F}
\newcommand{\intE}{\mathcal{O}_E}\begin{thm}
Let $E,\ F$ and $K$ be number fields in a tower:
$$K\subseteq F \subseteq E$$
and let $\intE,\ \intF$ and $\intK$ be their rings of integers respectively. Suppose $\p$ is a prime ideal of $\intK$ and let $\P$ be a prime ideal of $\intF$ lying above $\p$, and $\Pcal$ is a prime ideal of $\intE$ lying above $\P$.
\begin{center}
$\xymatrix{
{E} \ar@{-}[d] & {\intE} \ar@{-}[d] & {\Pcal} \ar@{-}[d] \\
{F} \ar@{-}[d] & {\intF} \ar@{-}[d] & {\P} \ar@{-}[d]\\
K & \intK & \p }$
\end{center}
Then the indices of the extensions, the ramification indices and inertial degrees satisfy:
\begin{eqnarray}
[E:K] &=& [E:F]\cdot [F:K],\\
\nonumber & &\\
e(\Pcal|\p) &=& e(\Pcal|\P)\cdot e(\P|\p),\\
\nonumber & &\\
f(\Pcal|\p) &=& f(\Pcal|\P)\cdot f(\P|\p).
\end{eqnarray}
\end{thm}