Joubert's theorem explained
In polynomial algebra and field theory, Joubert's theorem states that if
and
are
fields,
is a
separable field extension of
of
degree 6, and the
characteristic of
is not equal to 2, then
is generated over
by some element λ in
, such that the
minimal polynomial
of λ has the form
=
, for some constants
in
.
[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
is neither 2 nor 3).
[1] [8] In 2006 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(
) ≠ 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
and
.
[1] References
- Reichstein, Zinovy. Joubert's theorem fails in characteristic 2. Comptes Rendus Mathematique. 352. 10. 2014. 773–777. 10.1016/j.crma.2014.08.004 . 1406.7529. 1345373.
- Book: Société d'agriculture, sciences et arts de la Sarthe. Bulletin de la Société d'agriculture, sciences et arts de la Sarthe. 1895. Société d'agriculture, sciences et arts de la Sarthe. 16–.
- Book: Institut catholique de Paris. Le Livre Du Centenaire. 1976. Editions Beauchesne. 32.
- Web site: Joubert. cosmovisions.com.
- Goldstein, Catherine. Catherine Goldstein. Les autres de l'un: deux enquêtes prosopographiques sur Charles Hermite. 1209.5371. 2012. math.HO. (See footnote at bottom of page 18.)
- Book: Catalogue général de la librairie française: 1876-1885, auteurs : I-Z. 1887. Nilsson, P. Lamm. 29.
- Sur l'équation du sixième degré. Note du P. Joubert, présentée par M. Hermite. Comptes rendus hebdomadaires des séances de l'Académie des sciences. tome 64. Paris. Série A. 1835 . 1025–1029. (P. Joubert means le Père Joubert.)
- Coray. Daniel F.. Cubic hypersurfaces and a result of Hermite. Duke Mathematical Journal. 54. 2. 1987. 657–670. 0012-7094. 10.1215/S0012-7094-87-05428-7.
- Kraft, H.. A result of Hermite and equations of degree 5 and 6. J. Algebra. 297. 1. 2006. 234–253. 2206857. 10.1016/j.jalgebra.2005.04.015 . math/0403323. 8037344.