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) 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 field L of an n-dimensional algebraic variety over a ground field K is n.
- Q(√2, π) has transcendence degree 1 over Q because √2 is algebraic while π is transcendental.
- The transcendence degree of C or R over Q is the cardinality of the continuum. (This follows since any element has only countably many algebraic elements over it in Q, since Q is itself countable.)
- The transcendence degree of Q(π, e) over Q is either 1 or 2; the precise answer is unknown because it is not known whether π and e are algebraically independent.
Read more about this topic: Transcendence Degree
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