Perfect Fields

Manuel Eberl 📧 and Katharina Kreuzer 📧

November 6, 2023

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


This entry provides a type class for perfect fields. A perfect field K can be characterized by one of the following equivalent conditions:
  • Any irreducible polynomial p is separable, i.e. gcd(p,p') = 1, or, equivalently, p' non-zero.
  • Either the characteristic of K is 0 or p > 0 and the Frobenius endomorphism is surjective (i.e. every element of K has a p-th root).
We define perfect fields using the second characterization and show the equivalence to the first characterization. The implication ``2 => 1'' is relatively straightforward using the injectivity of the Frobenius homomorphism. Examples for perfect fields are:
  • any field of characteristic 0 (e.g. the reals or complex numbers)
  • any finite field (i.e. Fq for q=pn, n > 0 and p prime)
  • any algebraically closed field


BSD License


Session Perfect_Fields