Joubert's theorem

In polynomial algebra and field theory, Joubert's theorem states that if ${\displaystyle K}$ and ${\displaystyle L}$ are fields, ${\displaystyle L}$ is a separable field extension of ${\displaystyle K}$ of degree 6, and the characteristic of ${\displaystyle K}$ is not equal to 2, then ${\displaystyle L}$ is generated over ${\displaystyle K}$ by some element λ in ${\displaystyle L}$, such that the minimal polynomial ${\displaystyle p}$ of λ has the form ${\displaystyle p(t)}$ = ${\displaystyle t^{6}+c_{4}t^{4}+c_{2}t^{2}+c_{1}t+c_{0}}$, for some constants ${\displaystyle c_{4},c_{2},c_{1},c_{0}}$ in ${\displaystyle K}$.^[1] The theorem is named in honor of Charles Joubert, a French mathematician, lycée professor, and Jesuit priest.^{[2][3][4][5][6]}

In 1867 Joubert published his theorem in his paper Sur l'équation du sixième degré in tome 64 of Comptes rendus hebdomadaires des séances de l'Académie des sciences.^[7] He seems to have made the assumption that the fields involved in the theorem are subfields of the complex field.^[1]

Using arithmetic properties of hypersurfaces, Daniel F. Coray gave, in 1987, a proof of Joubert's theorem (with the assumption that the characteristic of ${\displaystyle K}$ is neither 2 nor 3).^[1][8] In 2006 Hanspeter Kraft [de] gave a proof of Joubert's theorem^[9] "based on an enhanced version of Joubert’s argument".^[1] In 2014 Zinovy Reichstein proved that the condition characteristic(${\displaystyle K}$) ≠ 2 is necessary in general to prove the theorem, but the theorem's conclusion can be proved in the characteristic 2 case with some additional assumptions on ${\displaystyle K}$ and ${\displaystyle L}$.^[1]