Joubert's theorem fails in characteristic 2

TL;DR

This paper demonstrates that Joubert's theorem, which describes a specific form of minimal polynomial for degree 6 extensions, does not hold in characteristic 2 fields, highlighting a limitation in the theorem's applicability.

Contribution

The paper proves that Joubert's theorem fails in characteristic 2, providing a counterexample and clarifying the theorem's limitations in this case.

Findings

Joubert's theorem does not hold in characteristic 2

Counterexamples exist in characteristic 2 fields

The minimal polynomial form is not guaranteed in characteristic 2

Abstract

Let L/K be a separable field extension of degree 6. An 1867 theorem of P. Joubert asserts that if char(K) is different from 2 then L is generated over K by an element whose minimal polynomial is of the form t^6 + a t^4 + b t^2 + ct + d for some a, b, c, d in K. We show that this theorem fails in characteristic 2.