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
Owner confidence rating: High Entry average rating: No information on entry rating
[parent] the compositum of a Galois extension and another extension is Galois (Theorem)
Theorem 1   Let $ E/K$ be a Galois extension of fields, let $ F/K$ be an arbitrary extension and assume that $ E$ and $ F$ are both subfields of some other larger field $ T$. The compositum of $ E$ and $ F$ is here denoted by $ EF$. Then:
  1. $ EF$ is a Galois extension of $ F$ and $ E$ is Galois over $ E\cap F$;
  2. Let $ H=\operatorname{Gal}(EF/F)$. The restriction map:
    $\displaystyle H=\operatorname{Gal}(EF/F)$ $\displaystyle \longrightarrow$ $\displaystyle \operatorname{Gal}(E/E\cap F)$  
    $\displaystyle \sigma$ $\displaystyle \longrightarrow$ $\displaystyle \sigma \vert _{E}$  

    is an isomorphism, where $ \sigma \vert _{E}$ denotes the restriction of $ \sigma$ to $ E$.
Remark 1   Notice, however, that if $ E/F$ and $ F/K$ are both Galois extensions, the extension $ E/K$ need not be Galois. See example of normal extension for a counterexample.



"the compositum of a Galois extension and another extension is Galois" is owned by alozano.
(view preamble)

View style:

See Also: fundamental theorem of Galois theory, Galois extension, example of normal extension, class number divisibility in extensions, Galois group of the compositum of two Galois extensions, extensions without unramified subextensions and class number divisibility

Keywords:  compositum, composite field, Galois extension

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

Cross-references: counterexample, example of normal extension, isomorphism, map, restriction, compositum, subfields, extension, fields, Galois extension
There is 1 reference to this entry.

This is version 3 of the compositum of a Galois extension and another extension is Galois, born on 2005-02-21, modified 2005-03-10.
Object id is 6791, canonical name is CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois.
Accessed 1310 times total.

Classification:
AMS MSC12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)
 11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory)

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)