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)
			(1)
$\displaystyle e(\mathcal{P}\|{\mathfrak{p}})$	$\displaystyle=$	$\displaystyle e(\mathcal{P}\|{\mathfrak{P}})\cdot e({\mathfrak{P}}\|{\mathfrak{p% }}),$	(2)
			(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
Canonical name	TheRamificationIndexAndTheInertialDegreeAreMultiplicativeInTowers
Date of creation	2013-03-22 15:06:34
Last modified on	2013-03-22 15:06:34
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	5
Author	alozano (2414)
Entry type	Theorem
Classification	msc 12F99
Classification	msc 13B02
Classification	msc 11S15
Related topic	Ramify
Related topic	InertialDegree