Abstract
For a finite group G and a field k,we call a G-Galois extension over k by G/k-extension. Whether a G/k-extension exists or not is the first version of inverse Galois problem. Especially the case when k = Q the rational number field, plays an important role in the study of the absolute Galois Group of Q. By many mathematicians, the existence of G/Q-extensions has been shown for a lot of finite groups G by now (cf. Malle-Matzat [14], Serre [19], etc.)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
The GAP Group, GAPâ Groups,Algorithms, and Programming, Version 4.3, 2002,http://www.gap-system.org.
W. Gröbner Minimalbasis der Quaternionengruppe, Monatshefte f. Math. and Physik 41 (1934), 78â84.
K. Hashimoto, Inverse Galois problems related to dihedral groups (Japanese), Algebraic number theory and related topics (Japanese) (Kyoto, 2000), SĂșrikaisekikenkyĂșsho KökyĂșroku No. 1154 (2000), 125â136.
K. Hashimoto, On Brumerâs family of RM-curves of genus two, Tohoku Math. J. 52 (2000), no. 4, 475â488.
K. Hashimoto and K. Miyake, Inverse Galois problem for dihedral groups, in âNumber theory and its applications (S. Kanemitsu and K. Gyory, ed.),â 165181, Developments in Mathematics, 2. Kluwer Acad. Publ., Doredrecht, 1999.
K. Hashimoto and Y. Rikuna, On generic families of cyclic polynomials with even degree, Manuscripta Math. 107 (2002), 283â288.
C. Jensen, A. Ledet, and N. Yui, âGeneric polynomials, constructive aspects of the inverse Galois problem,â Mathematical Sciences Research Institute Publications, Cambridge, 2003.
G. Kemper, Das Noethersche Problem and generische Polynome, dissertation, UniversitÀt Heidelberg, 1994.
G. Kemper, A consructive approach to Noetherâs problem, Manuscripta Math. 90 (1996), 343â363.
G. Kemper, Generic polynomials are descent-generic, Manuscripta Math. 105 (2001), 139â141.
G. Kemper and E. Mattig, Generic polynomials with few parameters, J. Symbolic Comp. 30 (2000), 843â857.
A. Ledet, Generic and explicit realisation of small p-groups, J. Symbolic Comp. 30 (2000), 859--865.
A. Ledet, Generic extensions and generic polynomials, J. Symbolic Comp. 30 (2000), 867â872.
G. Malle and B. H. Matzat, âInverse Galois theory,â Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999.
K. Miyake Linear Fractional Transformations and Cyclic Polynomials, Algebraic number theory (Hapcheon/Saga, 1996). Adv. Stud. Contemp. Math. (Pusan) 1, (1999), 137â142.
Y. Rikuna, On generic polynomials for the modular 2-groups, Proc. Japan Acad. Ser. A Mathe. Sci. 78 (2002, no. 3, 33â35.
Y. Rikuna, On simple families of cyclic polynomials, Proc. Amer. Math. Soc. 130 (2002), no 8, 2215â2218.
Y. Rikuna, On generic quasi-dihedral polynomials, preprint.
J.-P. Serre, âTopics in Galois Theory,â Research Notes in Mathematics, 1. Jones and Bartlett Publ., 1992.
D. ShanksThe simplest cubic fields, Math. Comp. 28 (1974), 1137â1152.
A. D. Thomas and G. V. Wood, âGroup tables,â Shiva Publishing Limited, 1980.
S. Wolfram, âThe Mathematica Book, 4th ed.,â Wolfram Media/Cambridge University Press, 1999.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Kluwer Academic Publishers
About this chapter
Cite this chapter
Rikuna, Y. (2004). Explicit Constructions of Generic Polynomials for Some Elementary Groups. In: Hashimoto, Ki., Miyake, K., Nakamura, H. (eds) Galois Theory and Modular Forms. Developments in Mathematics, vol 11. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-0249-0_9
Download citation
DOI: https://doi.org/10.1007/978-1-4613-0249-0_9
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-7960-7
Online ISBN: 978-1-4613-0249-0
eBook Packages: Springer Book Archive