## Families of cyclic polynomials obtained from geometric generalization of Gaussian period relations

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- by Ki-ichiro Hashimoto and Akinari Hoshi PDF
- Math. Comp.
**74**(2005), 1519-1530 Request permission

## Abstract:

A general method of constructing families of cyclic polynomials over $\mathbb {Q}$ with more than one parameter will be discussed, which may be called a geometric generalization of the Gaussian period relations. Using this, we obtain explicit multi-parametric families of cyclic polynomials over $\mathbb {Q}$ of degree $3\le e\le 7$. We also give a simple family of cyclic polynomials with one parameter in each case, by specializing our parameters.## References

- Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams,
*Gauss and Jacobi sums*, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1998. A Wiley-Interscience Publication. MR**1625181** - L.E. Dickson,
*Cyclotomy, higher congruences and Waring’s problem*, Amer. J. Math.**57**(1935), 391–424. - Carl Friedrich Gauss,
*Disquisitiones arithmeticae*, Yale University Press, New Haven, Conn.-London, 1966. Translated into English by Arthur A. Clarke, S. J. MR**0197380** - Marie-Nicole Gras,
*Special units in real cyclic sextic fields*, Math. Comp.**48**(1987), no. 177, 179–182. MR**866107**, DOI 10.1090/S0025-5718-1987-0866107-1 - K. Hashimoto and A. Hoshi
*Geometric generalization of Gaussian period relations with application to Noether’s problem for meta-cyclic groups*, to appear in Tokyo J. Math. - Christian U. Jensen, Arne Ledet, and Noriko Yui,
*Generic polynomials*, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, 2002. Constructive aspects of the inverse Galois problem. MR**1969648** - S. A. Katre and A. R. Rajwade,
*Complete solution of the cyclotomic problem in $\textbf {F}_q$ for any prime modulus $l,\;q=p^\alpha ,\;p\equiv 1\;(\textrm {mod}\,l)$*, Acta Arith.**45**(1985), no. 3, 183–199. MR**808019**, DOI 10.4064/aa-45-3-183-199 - D. H. Lehmer and Emma Lehmer,
*The Lehmer project*, Math. Comp.**61**(1993), no. 203, 313–317. MR**1189521**, DOI 10.1090/S0025-5718-1993-1189521-9 - Emma Lehmer,
*Connection between Gaussian periods and cyclic units*, Math. Comp.**50**(1988), no. 182, 535–541. MR**929551**, DOI 10.1090/S0025-5718-1988-0929551-0 - H. W. Lenstra Jr.,
*Rational functions invariant under a finite abelian group*, Invent. Math.**25**(1974), 299–325. MR**347788**, DOI 10.1007/BF01389732 - Gunter Malle and B. Heinrich Matzat,
*Inverse Galois theory*, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. MR**1711577**, DOI 10.1007/978-3-662-12123-8 - Katsuhiko Masuda,
*On a problem of Chevalley*, Nagoya Math. J.**8**(1955), 59–63. MR**69159** - Katsuhiko Masuda,
*Application of the theory of the group of classes of projective modules to the existance problem of independent parameters of invariant*, J. Math. Soc. Japan**20**(1968), 223–232. MR**223345**, DOI 10.2969/jmsj/02010223 - Emma Lehmer,
*Connection between Gaussian periods and cyclic units*, Math. Comp.**50**(1988), no. 182, 535–541. MR**929551**, DOI 10.1090/S0025-5718-1988-0929551-0 - Jean-Pierre Serre,
*Topics in Galois theory*, Research Notes in Mathematics, vol. 1, Jones and Bartlett Publishers, Boston, MA, 1992. Lecture notes prepared by Henri Damon [Henri Darmon]; With a foreword by Darmon and the author. MR**1162313** - Richard G. Swan,
*Invariant rational functions and a problem of Steenrod*, Invent. Math.**7**(1969), 148–158. MR**244215**, DOI 10.1007/BF01389798 - F. Thaine,
*Properties that characterize Gaussian periods and cyclotomic numbers*, Proc. Amer. Math. Soc.**124**(1996), no. 1, 35–45. MR**1301532**, DOI 10.1090/S0002-9939-96-03108-5 - F. Thaine,
*On the coefficients of Jacobi sums in prime cyclotomic fields*, Trans. Amer. Math. Soc.**351**(1999), no. 12, 4769–4790. MR**1475696**, DOI 10.1090/S0002-9947-99-02223-0 - F. Thaine,
*Families of irreducible polynomials of Gaussian periods and matrices of cyclotomic numbers*, Math. Comp.**69**(2000), no. 232, 1653–1666. MR**1653998**, DOI 10.1090/S0025-5718-99-01142-4 - F. Thaine,
*Jacobi sums and new families of irreducible polynomials of Gaussian periods*, Math. Comp.**70**(2001), no. 236, 1617–1640. MR**1836923**, DOI 10.1090/S0025-5718-01-01312-6 - F. Thaine,
*Cyclic polynomials and the multiplication matrices of their roots*, J. Pure Appl. Algebra**188**(2004), no. 1-3, 247–286. MR**2030817**, DOI 10.1016/j.jpaa.2003.07.004

## Additional Information

**Ki-ichiro Hashimoto**- Affiliation: Department of Mathematical Sciences, School of Science and Engineering, Waseda University, 3–4–1 Ohkubo, Shinjuku-ku, Tokyo 169–8555, Japan
- Email: khasimot@waseda.jp
**Akinari Hoshi**- Affiliation: Department of Mathematical Sciences, School of Science and Engineering, Waseda University, 3–4–1 Ohkubo, Shinjuku-ku, Tokyo 169–8555, Japan
- MR Author ID: 714371
- Email: hoshi@ruri.waseda.jp
- Received by editor(s): November 13, 2002
- Received by editor(s) in revised form: May 19, 2004
- Published electronically: February 14, 2005
- © Copyright 2005 American Mathematical Society
- Journal: Math. Comp.
**74**(2005), 1519-1530 - MSC (2000): Primary 11R18, 11R27, 11T22, 12F10, 12F12
- DOI: https://doi.org/10.1090/S0025-5718-05-01750-3
- MathSciNet review: 2137015