title: P-basis
text: # 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.
  