Topic summary
Transcendence degree
Extracted from the Wikipedia article Transcendental extension.
Examples
- An extension is algebraic if and only if its transcendence degree is 0; the empty set serves as a transcendence basis here.
- The field of rational functions in n variables K(x1,...,xn) (i.e. the field of fractions of the polynomial ringK[x1,...,xn]) is a purely transcendental extension with transcendence degree n over K; we can for example take {x1,...,xn} as a transcendence base.
- More generally, the transcendence degree of the function fieldL of an n-dimensional algebraic variety over a ground field K is n.
- Q(√2, e) has transcendence degree 1 over Q because √2 is algebraic while e is transcendental.
- The transcendence degree of C or R over Q is the cardinality of the continuum. (Since Q is countable, the field Q(S) will have the same cardinality as S for any infinite set S, and any algebraic extension of Q(S) will have the same cardinality again.)
- The transcendence degree of Q(e, π) over Q is either 1 or 2; the precise answer is unknown because it is not known whether e and π are algebraically independent (see Schanuel's conjecture).
- If S is a compactRiemann surface, the field C(S) of meromorphic functions on S has transcendence degree 1 over C.