Transcendence degree

The transcendence degree of a field extension L/K is the cardinality of any subset S of L such that the elements of S are algebraically independent over K and L is an algebraic extension of the field K(S) obtained by adjoining the elements of S to K. There is analogy with the theory dimension of vector spaces. The dictionary matches algebraically independent sets with linearly independent sets; sets S such that L is algebraic over K(S) with spanning sets; transcendence bases (sets S with both properties) with bases; and transcendence degree with dimension. The fact that transcendence bases always exist (like the fact that bases always exist in linear algebra) requires the axiom of choice. The proof that any two bases have the same cardinality depends, in each setting, on an exchange lemma.

If no field K is specified, the transcendence degree of a field L is its degree relative to the prime field of the same characteristic, i.e., Q if L is of characteristic 0 and F_p if L is of characteristic p.

Examples

• Every algebraic extension has transcendence degree 0.
• The field of rational functions in n variables K(x₁,...,x_n) has transcendence degree n over K.
• More generally, the transcendence degree of the function field L of an n-dimensional variety over a ground field K is n.
• The transcendence degree of C or R over Q is the cardinality of the continuum.