# the ramification index and the inertial degree are multiplicative in towers

###### Theorem.

Let $E,\ F$ and $K$ be number fields in a tower:

 $K\subseteq F\subseteq E$

and let $\mathcal{O}_{E},\ \mathcal{O}_{F}$ and $\mathcal{O}_{K}$ be their rings of integers respectively. Suppose ${\mathfrak{p}}$ is a prime ideal of $\mathcal{O}_{K}$ and let ${\mathfrak{P}}$ be a prime ideal of $\mathcal{O}_{F}$ lying above ${\mathfrak{p}}$, and $\mathcal{P}$ is a prime ideal of $\mathcal{O}_{E}$ lying above ${\mathfrak{P}}$.

$\xymatrix{{E}\ar@{-}[d]&{\mathcal{O}_{E}}\ar@{-}[d]&{\mathcal{P}}\ar@{-}[d]% \inner@par{F}\ar@{-}[d]&{\mathcal{O}_{F}}\ar@{-}[d]&{{\mathfrak{P}}}\ar@{-}[d]% \inner@par K&\mathcal{O}_{K}&{\mathfrak{p}}}$

Then the indices of the extensions, the ramification indices and inertial degrees satisfy:

 $\displaystyle[E:K]$ $\displaystyle=$ $\displaystyle[E:F]\cdot[F:K],$ (1) $\displaystyle e(\mathcal{P}|{\mathfrak{p}})$ $\displaystyle=$ $\displaystyle e(\mathcal{P}|{\mathfrak{P}})\cdot e({\mathfrak{P}}|{\mathfrak{p% }}),$ (2) $\displaystyle f(\mathcal{P}|{\mathfrak{p}})$ $\displaystyle=$ $\displaystyle f(\mathcal{P}|{\mathfrak{P}})\cdot f({\mathfrak{P}}|{\mathfrak{p% }}).$ (3)
Title the ramification index and the inertial degree are multiplicative in towers TheRamificationIndexAndTheInertialDegreeAreMultiplicativeInTowers 2013-03-22 15:06:34 2013-03-22 15:06:34 alozano (2414) alozano (2414) 5 alozano (2414) Theorem msc 12F99 msc 13B02 msc 11S15 Ramify InertialDegree