Abstract
Let k be an arbitrary field. We study a general method to solve the subfield problem of generic polynomials for the symmetric groups over k via Tschirnhausen transformation. Based on the general result in the former part, we give an explicit solution to the field isomorphism problem and the subfield problem of cubic generic polynomials for \( \mathfrak{S} \)3 and C3 over k. As an application of the cubic case, we also give several sextic generic polynomials over k.
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References
E. R. Berlekamp, An analog to the discriminant over fields of characteristic two, J. Algebra 38 (1976), 315–317.
A. A. Bruen, C. U. Jensen, N. Yui, Polynomials with Frobenius groups of prime degree as Galois Groups II, J. Number Theory 24 (1986), 305–359.
J. Buhler, Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (rd), 159–179.
R. J. Chapman, Automorphism polynomials in cyclic cubic extensions, J. Number Theory 61 (1996), 283–291.
H. Chu, S. Hu, M. Kang, J. Zhang, Groups with essential dimension one, preprint. arXiv:math/0611917vl [math.AG].
H. Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, 138. Springer-Verlag, Berlin, 1993.
K. Hashimoto, A. Hoshi, Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations, Math. Comp. 74 (2005), 1519–1530.
K. Hashimoto, A. Hoshi, Geometric generalization of Gaussian period relations with application to Noether’s problem for meta-cyclic groups, Tokyo J. Math. 28 (2005), 13–32.
K. Hashimoto, K. Miyake, Inverse Galois problem for dihedral groups, Number theory and its applications, Dev. Math., 2, Doredrecht: Kluwer Acad. Publ. 165–181, 1999.
M. Hindry, J. H. Silverman, Diophantine geometry. An introduction, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, 2000.
A. Hoshi, K. Miyake, Tschirnhausen transformation of a cubic generic polynomial and a 2-dimensional involutive Cremona transformation, Proc. Japan Acad., Series A 83 (2007), 21–26.
B. Huppert, Endliche Gruppen. I., Grundlehren der Mathematischen Wissenschaften 134, Springer-Verlag, Berlin-New York 1967.
C. Jensen, A. Ledet, N. Yui, Generic polynomials, constructive aspects of the inverse Galois problem, Mathematical Sciences Research Institute Publications, Cambridge, 2002.
G. Kemper, A constructive approach to Noether’s problem, Manuscripta Math. 90 (1996), 343–363.
G. Kemper, Generic polynomials are descent-generic, Manuscripta Math. 105 (2001), 139–141.
G. Kemper, E. Mattig, Generic polynomials with few parameters, J. Symbolic Comput. 30 (2000), 843–857.
M. Kida, Kummer theory for norm algebraic tori, J. Algebra 293 (2005), 427–447.
M. Kida, Cyclic polynomials arising from Kummer theory of norm algebraic tori, Algorithmic number theory, Lecture Notes in Comput. Sci., 4076, Springer, Berlin, 102–113, 2006.
T. Komatsu, Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummer theory, Manuscripta math. 114 (2004), 265–279.
T. Komatsu, Generic sextic polynomial related to the subfield problem of a cubic polynomial, preprint. http://www.math.kyushu-u.ac.jp/coe/report/pdf/2006–9.pdf
S. Lang, Elliptic curves: Diophantine analysis, Grundlehren der Mathematischen Wissenschaften 231, Springer, 1978.
S. Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983.
A. Ledet, On groups with essential dimension one, J. Algebra, 311 (2007), 31–37.
K. Miyake, Linear fractional transformations and cyclic polynomials, Algebraic number theory (Hapcheon/Saga, 1996). Adv. Stud. Contemp. Math. (Pusan) 1 (1999), 137–142.
K. Miyake, Some families of Mordell curves associated to cubic fields, J. Comput. Appl. Math. 160 (2003), 217–231.
K. Miyake, An introduction to elliptic curves and their Diophantine geometry—Mordell curves, Ann. Sci. Math. Québec 28 (2004), 165–178.
K. Miyake, Cubic fields and Mordell curves, Number theory, 175–183, Dev. Math., 15, Springer, New York, 2006.
P. Morton, Characterizing cyclic cubic extensions by automorphism polynomials, J. Number Theory 49 (1994), 183–208.
D. Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152.
S. Wolfram, The Mathematica Book, Fifth Edition, Wolfram Media, Inc., Cambridge University Press, 2003.
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Hoshi, A., Miyake, K. (2009). A Geometric Framework for the Subfield Problem of Generic Polynomials Via Tschirnhausen Transformation. In: Adhikari, S.D., Ramakrishnan, B. (eds) Number Theory and Applications. Hindustan Book Agency, Gurgaon. https://doi.org/10.1007/978-93-86279-46-0_7
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DOI: https://doi.org/10.1007/978-93-86279-46-0_7
Publisher Name: Hindustan Book Agency, Gurgaon
Print ISBN: 978-81-85931-97-5
Online ISBN: 978-93-86279-46-0
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