Joubert's theorem

Last updated November 29, 2024

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 $K$ .^[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 $K$ 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( $K$ ) ≠ 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 $L$ .^[1]

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References

1 2 3 4 5 Reichstein, Zinovy (2014). "Joubert's theorem fails in characteristic 2". Comptes Rendus Mathematique. 352 (10): 773–777. arXiv: 1406.7529 . Bibcode:2014CRMat.352..773R. 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. tome 64. Paris: 1025–1029. 1835. (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.

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[Reichstein-1] 1 2 3 4 5 Reichstein, Zinovy (2014). "Joubert's theorem fails in characteristic 2". Comptes Rendus Mathematique. 352 (10): 773–777. arXiv: 1406.7529 . Bibcode:2014CRMat.352..773R. doi:10.1016/j.crma.2014.08.004. S2CID 1345373.

[Sarthe1895-2] 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–.

[Paris1976-3] Institut catholique de Paris (1976). Le Livre Du Centenaire. Editions Beauchesne. p. 32.

[4] "Joubert". cosmovisions.com.

[5] 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.)

[6] Catalogue général de la librairie française: 1876-1885, auteurs : I-Z. Nilsson, P. Lamm. 1887. p. 29.

[7] "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. tome 64. Paris: 1025–1029. 1835. (P. Joubert means le Père Joubert.)

[Coray1987-8] 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.

[9] 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.

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