Transcendence Degree

The transcendence degree of Q(pi), sometimes called the transcendental degree, is one because it is generated by one extra element. In contrast, Q(pi,pi^2) (which is the same field) also has transcendence degree one because pi^2 is algebraic over Q(pi). 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.

For instance, the transcendence degree of Q(sqrt(2),pi) over Q is one. The transcendence degree of R over Q is an infinite cardinal number. There are many open questions about the traditional constants in mathematics, such as the transcendence degree of Q(e,pi).

See also

Algebraic Extension, Extension Field, Extension Field Degree, Field, Irrationality Measure, Transcendental Extension

This entry contributed by Todd Rowland

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Rowland, Todd. "Transcendence Degree." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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