On the essential dimension of cyclic p-groups THANKSREF="*" ID="*"The author gratefully acknowledges support from the Swiss National Science Fundation, grant no. 200020-109174/1 (project leader: E. Bayer-Fluckiger)
Tóm tắt
Let p be a prime number, let K be a field of characteristic not p, containing the p-th roots of unity, and let r≥1 be an integer. We compute the essential dimension of ℤ/p
r
ℤ over K (Theorem 4.1). In particular, i) We have edℚ(ℤ/8ℤ)=4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished). ii) We have edℚ(ℤ/p
r
ℤ)≥p
r-1.
Tài liệu tham khảo
Berhuy, G., Favi, G.: Essential dimension: a functorial point of view (after A. Merkurjev). Doc. Math. 8, 279–330 (2003)
Brauer, R.: Über den Index und den Exponenten von Divisionsalgebren. Tohoku Math. J. 37, 77–87 (1933)
Buhler, J., Reichstein, Z.: On the essential dimension of a finite group. Compos. Math. 106, 159–179 (1997)
Favi, G., Florence, M.: Tori and Essential Dimension. To appear in J. Algebra, available at http://www.math.uni-bielefeld.de/LAG/man/208.pdf
Jensen, C.U., Ledet, A., Yui, N.: Generic Polynomials: Constructive Aspects of the Inverse Galois Problem. Math. Sci. Res. Inst. Publ. Ser., vol. 45. Cambridge University Press (2002)
Karpenko, N.: On anisotropy of orthogonal involutions. J. Ramanujan Math. Soc. 15(1), 1–22 (2002)
Ledet, A.: On the essential dimension of some semi-direct products. Can. Math. Bull. 45, 422–427 (2002)
Merkurjev, A.: Degree formula. Available at http://www.mathematik.uni-bielefeld.de/rost/degree-formula.html
Reichstein, Z.: On the notion of essential dimension for algebraic groups. Transform. Groups 5(3), 265–304 (2000)
Rost, M.: Essential dimension of twisted C 4. Available at http://www.mathematik.uni-bielefeld.de/rost/ed.html#C4
Rowen, L.: Ring Theory, vol. 2. Pure Appl. Math., vol. 128. Academic Press, Boston (1988)
Serre, J.-P.: Groupes algébriques et corps de classes. Publ. Math. Inst. Math. Univ. Nancago VII. Hermann (1959)