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the ramification index and the inertial degree are multiplicative in towers
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(Theorem)
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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 $\P$ .
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}
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"the ramification index and the inertial degree are multiplicative in towers" is owned by alozano.
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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 MSC: | 11S15 (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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Pending Errata and Addenda
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![$ \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}}}$](http://images.planetmath.org:8080/cache/objects/6842/js/img1.png "$ \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}}}$")