transcendental degree

The transcendence degree of Q(π), sometimes called the transcendental degree, is one because it is generated by one extra element. In contrast, Q(π, π^2) (which is the same field) also has transcendence degree one because π^2 is algebraic over Q(π). In general, the transcendence degree of an extension field K over a field F is the smallest number elements of K which are not algebraic over F, but needed to generate K. If the smallest set of transcendental elements needed to generate K is infinite, then the transcendence degree is the cardinal number of that set.

algebraic extension | extension field | extension field degree | field | irrationality measure | transcendental extension