On derivations of fields of algebraic functions

In this note, we shall prove that the transcendency degree of a finitely generated field extension is equal to a certain integer associated with a restriction mapping of spaces of derivations.

The Automorphisms of an Algebraically Closed Field

It is well known that the complex number field has infinitely many automorphisms. Moreover, it seems to be part of the folklore that the family of all automorphisms of the complex field has cardinality 2c, where c = 2ℵo. In this article the following generalization of this fact is proved: If k is any algebraically closed field then the family of all automorphisms of k has cardinality 2card k.The complex field has infinite transcendency degree over its prime subfield. For fields of this type the proof is accomplished by essentially permuting the elements in a transcendency basis and extending each permutation to an automorphism of the field.