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 (194)
Orphanage
Unclass'd
Unproven (498)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] simple transcendental field extension (Corollary)

The extension field $ K(\alpha)$ of a base field $ K$, where $ \alpha$ is transcendental with respect to $ K$, is a simple transcendental extension of $ K$. All such extension fields are isomorphic to the field $ K(X)$ of rational functions in one indeterminate $ X$ over $ K$, and thus to each other.

Example. The subfields $ \mathbb{Q}(\pi)$ and $ \mathbb{Q}(e)$ of $ \mathbb{R}$, where $ \pi$ is Ludolph's constant and $ e$ Napier's constant, are isomorphic.



"simple transcendental field extension" is owned by pahio.
(view preamble)

View style:

See Also: function field

Other names:  simple infinite field extension

This object's parent.

Attachments:
non-constant element of rational function field (Theorem) by pahio
Log in to rate this entry.
(view current ratings)

Cross-references: Napier's constant, subfields, indeterminate, rational functions, field, isomorphic, transcendental extension, transcendental, base field, extension field
There are 2 references to this entry.

This is version 7 of simple transcendental field extension, born on 2005-02-15, modified 2006-06-04.
Object id is 6751, canonical name is SimpleTranscendentalFieldExtension.
Accessed 2222 times total.

Classification:
AMS MSC: 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 | add example | add (any)