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[parent] the ramification index and the inertial degree are multiplicative in towers (Theorem)
Theorem 1   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 $\P$ .
$ \xymatrix{ {E} \ar@{-}[d] & {\mathcal{O}_E} \ar@{-}[d] & {\mathcal{P}} \ar@{-}... ...thcal{O}_F} \ar@{-}[d] & {\P } \ar@{-}[d]\ K & \mathcal{O}_K& {\mathfrak{p}}}$

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}




"the ramification index and the inertial degree are multiplicative in towers" is owned by alozano.
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See Also: ramification index, inertial degree

Keywords:  towers of number fields, ramification, inertia

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Cross-references: inertial degrees, ramification, extensions, indices, prime ideal, rings of integers, number fields

This is version 2 of the ramification index and the inertial degree are multiplicative in towers, born on 2005-03-03, modified 2005-03-03.
Object id is 6842, canonical name is RamificationIndexAndTheInertialDegreeAreMultiplicativeInTowers.
Accessed 1486 times total.

Classification:
AMS MSC11S15 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Ramification and extension theory)
 13B02 (Commutative rings and algebras :: Ring extensions and related topics :: Extension theory)
 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)

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