p-basis

In algebra, a p-basis is a generalization of the notion of a separating transcendence basis for a field extension of characteristic p, introduced by Teichmüller (1936).

Definition

Suppose k is a field of characteristic p and K is a field extension. A p-basis is a set of elements x_i of K such that the elements dx_i form a basis for the K-vector space Ω_K/k of differentials.

Examples

If K is a finitely generated separable extension of k then a p-basis is the same as a separating transcendence basis. In particular in this case the number of elements of the p-basis is the transcendence degree.
If k is a field, x an indeterminate, and K the field generated by all elements x^1/pⁿ then the empty set is a p-basis, though the extension is separable and has transcendence degree 1.
If K is a degree p extension of k obtained by adjoining a pth root t of an element of k then t is a p-basis, so a p-basis has cardinality 1 while the transcendence degree is 0.

References

Mac Lane, Saunders (1939), "Modular fields. I. Separating transcendence bases", Duke Math. J., 5 (2): 372–393, doi:10.1215/S0012-7094-39-00532-6, MR 1546131
Teichmüller, O. (1936), "p-Algebren", Deutsche Mathematik, 1: 362–388