The construction of solvable polynomials
HTML articles powered by AMS MathViewer
- by Harold M. Edwards PDF
- Bull. Amer. Math. Soc. 46 (2009), 397-411 Request permission
Erratum: Bull. Amer. Math. Soc. 46 (2009), 703-704.
Abstract:
Although Leopold Kronecker’s 1853 paper “On equations that are algebraically solvable” is famous for containing his enunciation of the Kronecker-Weber theorem, its main theorem is an altogether different one, a theorem that reduces the problem of constructing solvable polynomials of prime degree $\mu$ to the problem of constructing cyclic polynomials of degree $\mu -1$. Kronecker’s statement of the theorem is sketchy, and he gives no proof at all. There seem to have been very few later treatments of the theorem, none of them very clear and none more recent than 1924. A corrected version and a full proof of the theorem are given. The main technique is a constructive version of Galois theory close to Galois’s own.References
- A2 N. H. Abel, Extraits de quelques lettres à Crelle, Oeuvres, vol. 2, p. 266 (of the 1881 edition).
- David A. Cox, Galois theory, Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2004. MR 2119052, DOI 10.1002/9781118033081
- Harold M. Edwards, On the Kronecker Nachlass, Historia Math. 5 (1978), no. 4, 419–426 (English, with French summary). MR 511178, DOI 10.1016/0315-0860(78)90210-0
- Harold M. Edwards, Galois theory, Graduate Texts in Mathematics, vol. 101, Springer-Verlag, New York, 1984. MR 743418
- Harold M. Edwards, An appreciation of Kronecker, Math. Intelligencer 9 (1987), no. 1, 28–35. MR 869537, DOI 10.1007/BF03023570
- Harold M. Edwards, Essays in constructive mathematics, Springer-Verlag, New York, 2005. MR 2104015
- Harold M. Edwards, Kronecker’s fundamental theorem of general arithmetic, Episodes in the history of modern algebra (1800–1950), Hist. Math., vol. 32, Amer. Math. Soc., Providence, RI, 2007, pp. 107–116. MR 2353493, DOI 10.1090/hmath/032/06 F R. Fricke, Lehrbuch der Algebra, Vieweg, Braunschweig, 1924. G É. Galois, “Mémoire sur les conditions de résolubilité des équations par radicaux” in Écrits et Mémoires mathématiques, Paris, 1976, pp. 43-101 (English translation in [4], pp. 101-113).
- Leopold Kronecker, Leopold Kronecker’s Werke. Bände I–V, Chelsea Publishing Co., New York, 1968 (German). Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften von K. Hensel. MR 0237286
- Leopold Kronecker, Leopold Kronecker’s Werke. Bände I–V, Chelsea Publishing Co., New York, 1968 (German). Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften von K. Hensel. MR 0237286
- Leopold Kronecker, Leopold Kronecker’s Werke. Bände I–V, Chelsea Publishing Co., New York, 1968 (German). Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften von K. Hensel. MR 0237286
- Leopold Kronecker, Leopold Kronecker’s Werke. Bände I–V, Chelsea Publishing Co., New York, 1968 (German). Herausgegeben auf Veranlassung der Königlich Preussischen Akademie der Wissenschaften von K. Hensel. MR 0237286 N E. Netto, Theory of Substitutions (a translation, with extensive revisions by the author, of an 1880 work Substitutionentheorie), Wahr, Ann Arbor, 1892. (Chelsea reprint, 1964). N1 E. Netto, Vorlesungen über Algebra, Teubner, Leipzig, 1900.
- Birgit Petri and Norbert Schappacher, From Abel to Kronecker: episodes from 19th century algebra, The legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 227–266. MR 2077575
- H. Weber, Theorie der Abel’schen Zahlkörper, Acta Math. 8 (1886), no. 1, 193–263 (German). MR 1554698, DOI 10.1007/BF02417089 W1 H. Weber, Lehrbuch der Algebra, Vieweg, Braunschweig, 1895 (Reprint, AMS/Chelsea).
- A. Wiman, Über die metacyklischen Gleichungen von Primzahlgrad, Acta Math. 27 (1903), no. 1, 163–175 (German). MR 1554979, DOI 10.1007/BF02421303
Additional Information
- Harold M. Edwards
- Affiliation: Department of Mathematics, New York University, 251 Mercer St., New York, New York 10012
- Received by editor(s): November 21, 2008
- Received by editor(s) in revised form: January 13, 2009
- Published electronically: March 26, 2009
- © Copyright 2009 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 46 (2009), 397-411
- MSC (2000): Primary 11R32, 11R37, 11R18
- DOI: https://doi.org/10.1090/S0273-0979-09-01253-1
- MathSciNet review: 2507276