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)