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 (197)
Orphanage
Unclass'd
Unproven (498)
Corrections (22)
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
transcendence degree (Definition)

The transcendence degree of a set $ S$ over a field $ K$, denoted $ T_S$, is the size of the maximal subset $ S'$ of $ S$ such that all the elements of $ S'$ are algebraically independent.

The transcendence degree of a field extension $ L$ over $ K$ is the transcendence degree of the minimal subset of $ L$ needed to generate $ L$ over $ K$.

Heuristically speaking, the transcendence degree of a finite set $ S$ is obtained by taking the number of elements in the set, subtracting the number of algebraic elements in that set, and then subtracting the number of algebraic relations between distinct pairs of elements in $ S$.

Example 1 (Computing the Transcendence Degree)   The set $ S=\{\sqrt{7}, \pi, \pi^2, e\}$ has transcendence $ T_S\leq2$ over $ \mathbb{Q}$ since there are four elements, $ \sqrt{7}$ is algebraic, and the polynomial $ f(x,y)=x^2-y$ gives an algebraic dependence between $ \pi$ and $ \pi^2$ (i.e. $ (\pi,\pi^2)$ is a root of $ f$), giving $ T_S\leq4-1-1=2$. If we assume the conjecture that $ e$ and $ \pi$ are algebraically independent, then no more dependencies can exist, and we can conclude that, in fact, $ T_S=2$.



"transcendence degree" is owned by mathcam. [ full author list (2) ]
(view preamble)

View style:

Also defines:  transcendence degree of a set, transcendence degree of a field extension
Log in to rate this entry.
(view current ratings)

Cross-references: conjecture, root, algebraic dependence, polynomial, relations, algebraic, number, finite set, generate, minimal, field extension, algebraically independent, subset, size, field
There are 2 references to this entry.

This is version 4 of transcendence degree, born on 2003-09-25, modified 2007-05-09.
Object id is 4740, canonical name is TransendenceDegree.
Accessed 3972 times total.

Classification:
AMS MSC: 12F20 (Field theory and polynomials :: Field extensions :: Transcendental extensions)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy
"size" by polarbear on 2007-04-20 12:51:23
The word size sends to an object from graph theory which is not what you mean. Can you modify it somehow?
Bogdan
[ reply | up ]
Interact
post | correct | update request | add derivation | add example | add (any)