transcendence degree

The transcendence degreeMathworldPlanetmath of a set S over a field K, denoted TS, is the size of the maximal subset S of S such that all the elements of S are algebraically independentMathworldPlanetmath.

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

Heuristically speaking, the transcendence degree of a finite setMathworldPlanetmath 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 relationsMathworldPlanetmath between distinct pairs of elements in S.

Example 1 (Computing the Transcendence Degree).

The set S={7,π,π2,e} has transcendence TS2 over Q since there are four elements, 7 is algebraic, and the polynomialPlanetmathPlanetmath f(x,y)=x2-y gives an algebraic dependence between π and π2 (i.e. (π,π2) is a root of f), giving TS4-1-1=2. If we assume the conjecture that e and π are algebraically independent, then no more dependencies can exist, and we can conclude that, in fact, TS=2.

Title transcendence degree
Canonical name TranscendenceDegree
Date of creation 2013-03-22 13:58:11
Last modified on 2013-03-22 13:58:11
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 7
Author mathcam (2727)
Entry type Definition
Classification msc 12F20
Defines transcendence degree of a set
Defines transcendence degree of a field extension