Heuristically speaking, the transcendence degree of a finite set 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 .
Example 1 (Computing the Transcendence Degree).
The set has transcendence over since there are four elements, is algebraic, and the polynomial gives an algebraic dependence between and (i.e. is a root of ), giving . If we assume the conjecture that and are algebraically independent, then no more dependencies can exist, and we can conclude that, in fact, .
|Date of creation||2013-03-22 13:58:11|
|Last modified on||2013-03-22 13:58:11|
|Last modified by||mathcam (2727)|
|Defines||transcendence degree of a set|
|Defines||transcendence degree of a field extension|