# Joubert's theorem

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] References ^ Jump up to: a b c d e Reichstein, Zinovy (2014). "Joubert's theorem fails in characteristic 2". Comptes Rendus Mathematique. 352 (10): 773–777. arXiv:1406.7529. doi:10.1016/j.crma.2014.08.004. S2CID 1345373. ^ Société d'agriculture, sciences et arts de la Sarthe (1895). Bulletin de la Société d'agriculture, sciences et arts de la Sarthe. Société d'agriculture, sciences et arts de la Sarthe. pp. 16–. ^ Institut catholique de Paris (1976). Le Livre Du Centenaire. Editions Beauchesne. p. 32. ^ "Joubert". cosmovisions.com. ^ Goldstein, Catherine (2012). "Les autres de l'un: deux enquêtes prosopographiques sur Charles Hermite". arXiv:1209.5371 [math.HO]. (See footnote at bottom of page 18.) ^ Catalogue général de la librairie française: 1876-1885, auteurs : I-Z. Nilsson, P. Lamm. 1887. p. 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. Série A. Paris. tome 64: 1025–1029. (P. Joubert means le Père Joubert.) ^ Coray, Daniel F. (1987). "Cubic hypersurfaces and a result of Hermite". Duke Mathematical Journal. 54 (2): 657–670. doi:10.1215/S0012-7094-87-05428-7. ISSN 0012-7094. ^ Kraft, H. (2006). "A result of Hermite and equations of degree 5 and 6". J. Algebra. 297 (1): 234–253. arXiv:math/0403323. doi:10.1016/j.jalgebra.2005.04.015. MR 2206857. S2CID 8037344. Categories: Field (mathematics)Theorems in abstract algebra

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