PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
[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:
$\displaystyle K\subseteq F \subseteq E$
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 $ \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 $ \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:

$\displaystyle [E:K]$ $\displaystyle =$ $\displaystyle [E:F]\cdot [F:K],$ (1)
       
$\displaystyle e(\mathcal{P}\vert{\mathfrak{p}})$ $\displaystyle =$ $\displaystyle e(\mathcal{P}\vert\P )\cdot e(\P \vert{\mathfrak{p}}),$ (2)
       
$\displaystyle f(\mathcal{P}\vert{\mathfrak{p}})$ $\displaystyle =$ $\displaystyle f(\mathcal{P}\vert\P )\cdot f(\P \vert{\mathfrak{p}}).$ (3)



"the ramification index and the inertial degree are multiplicative in towers" is owned by alozano.
(view preamble)

View style:

See Also: ramification index, inertial degree

Keywords:  towers of number fields, ramification, inertia

This object's parent.
Log in to rate this entry.
(view current ratings)

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 1087 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)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)