Joubert's theorem

In polynomial algebra and field theory, Joubert's theorem states that if $K$ and $L$ are fields, $L$ is a separable field extension of $K$ of degree 6, and the characteristic of $K$ is not equal to 2, then $L$ is generated over $K$ by some element λ in $L$, such that the minimal polynomial $p$ of λ has the form $p(t)$ = $t^6\; +\; c\_4\; t^4\; +\; c\_2\; t^2\; +\; c\_1\; t\; +\; c\_0$, for some constants $c\_4,\; c\_2,\; c\_1,\; c\_0$ in $K$.{{cite journal|author=Reichstein, Zinovy|title=Joubert's theorem fails in characteristic 2|journal=Comptes Rendus Mathematique|volume=352|issue=10|year=2014|pages=773–777|doi=10.1016/j.crma.2014.08.004 |arxiv=1406.7529|bibcode=2014CRMat.352..773R |s2cid=1345373}} The theorem is named in honor of Charles Joubert, a French mathematician, lycée professor, and Jesuit priest.{{cite book|author=Société d'agriculture, sciences et arts de la Sarthe|title=Bulletin de la Société d'agriculture, sciences et arts de la Sarthe|url=https://books.google.com/books?id=M8hlCrY9218C&pg=PA16|year=1895|publisher=Société d'agriculture, sciences et arts de la Sarthe|pages=16–}}{{cite book|author=Institut catholique de Paris|title=Le Livre Du Centenaire|url=https://books.google.com/books?id=UHnK3KLjJYMC&pg=PA32|year=1976|publisher=Editions Beauchesne|page=32}}{{cite web|title=Joubert|website=cosmovisions.com|url=http://www.cosmovisions.com/Joubert.htm}}{{cite arXiv|author=Goldstein, Catherine|author-link=Catherine Goldstein|title=Les autres de l'un: deux enquêtes prosopographiques sur Charles Hermite|eprint=1209.5371|year=2012|class=math.HO}} (See footnote at bottom of page 18.){{cite book|title=Catalogue général de la librairie française: 1876-1885, auteurs : I-Z|url=https://books.google.com/books?id=Mdm2ahoa_mEC&pg=PA29|year=1887|publisher=Nilsson, P. Lamm|page=29}}

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.{{cite journal|title=Sur l'équation du sixième degré. Note du P. Joubert, présentée par M. Hermite|journal=Comptes rendus hebdomadaires des séances de l'Académie des sciences|volume=tome 64|location=Paris|series=Série A|year=1835 |pages=1025–1029|url=

https://babel.hathitrust.org/cgi/pt?id=umn.31951d000083952&view=1up&seq=1033&q1=joubert}} (P. Joubert means le Père Joubert.) He seems to have made the assumption that the fields involved in the theorem are subfields of the complex field.

Using arithmetic properties of hypersurfaces, Daniel F. Coray gave, in 1987, a proof of Joubert's theorem (with the assumption that the characteristic of $K$ is neither 2 nor 3).{{cite journal|last1=Coray|first1=Daniel F.|title=Cubic hypersurfaces and a result of Hermite|journal=Duke Mathematical Journal|volume=54|issue=2|year=1987|pages=657–670|issn=0012-7094|doi=10.1215/S0012-7094-87-05428-7}} In 2006 {{ill|Hanspeter Kraft|de}} gave a proof of Joubert's theorem{{cite journal|author=Kraft, H.|title=A result of Hermite and equations of degree 5 and 6|journal=J. Algebra|volume=297|issue=1|year=2006|pages=234–253|mr= 2206857|doi=10.1016/j.jalgebra.2005.04.015 |arxiv=math/0403323|s2cid=8037344}} "based on an enhanced version of Joubert’s argument". In 2014 Zinovy Reichstein proved that the condition characteristic($K$) ≠ 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 $K$ and $L$.