### A construction of quintic rings

Anthony C. Kable and Akihiko Yukie
Source: Nagoya Math. J. Volume 173 (2004), 163-203.

#### Abstract

We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant.

First Page:
Primary Subjects: 11R21
Secondary Subjects: 11E76, 11R27, 11S90
Full-text: Open access

Permanent link to this document: http://projecteuclid.org/euclid.nmj/1114631987
Mathematical Reviews number (MathSciNet): MR2041760
Zentralblatt MATH identifier: 02111503

### References

K. Amano, M. Fujigami and T. Kogiso, Construction of Irreducible Relative Invariant of the Prehomogeneous Vector Space $(SL_5 \times GL_4, \La^2(\mathbbC^5) \otimes \mathbbC^4)$ , Linear Algebra Appl., 355 (2002), 215–222.
Mathematical Reviews (MathSciNet): MR1930146
Digital Object Identifier: doi:10.1016/S0024-3795(02)00348-8
Zentralblatt MATH: 1027.11091
M. Bhargava, Higher Composition Laws, Ph.D. Thesis, Princeton University (2001).
Mathematical Reviews (MathSciNet): MR2702004
M. Bhargava, Gauss Composition and Generalizations , 1–8, Algorithmic Number Theory: 5th International Symposium (C. Fieker and D. R. Kohel, eds.), Springer Lecture Notes in Computer Science, vol. 2369, Springer, New York (2002).
Mathematical Reviews (MathSciNet): MR2041069
Zentralblatt MATH: 1058.11030
A. Cayley, On the theory of linear transformations , 80–94, The Collected Mathematical Papers of Arthur Cayley, vol. 1, Cambridge University Press, Cambridge (1889).
H. Cohen, F. Diaz y Diaz and M. Olivier, Counting discriminants of number fields of degree up to four , 269–283, Algorithmic Number Theory (Leiden 2000), Lecture Notes in Computer Science, vol. 1838, Springer, Berlin (2000).
Mathematical Reviews (MathSciNet): MR1850611
Zentralblatt MATH: 0987.11080
C. W. Curtis and I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Integral Orders, Vol. 1, Wiley Classics Library, John Wiley & Sons, New York (1990).
Mathematical Reviews (MathSciNet): MR1038525
H. Davenport and H. Heilbronn, On the density of discriminants of cubic fields , Bull. Lond. Math. Soc., 1 (1969), 345–348.
Mathematical Reviews (MathSciNet): MR254010
H. Davenport and H. Heilbronn, On the density of the discriminants of cubic fields. II , Proc. Roy. Soc. London Ser. A. 322 (1971), 405–420.
Mathematical Reviews (MathSciNet): MR491593
B. N. Delone and D. K. Fadeev, Theory of Irrationalities of the Third Degree, Translations of Mathematical Monographs, Vol. 10, American Mathematical Society, Providence (1964).
Mathematical Reviews (MathSciNet): MR160744
Zentralblatt MATH: 0133.30202
J. Dieudonné, Sur la reduction canonique des couples de matrices , Bull. Soc. math. France, 74 (1946), 130–146.
Mathematical Reviews (MathSciNet): MR22826
I. M. Gel'fand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Mathematics: Theory and Applications, Birkhäuser, Boston (1994).
Mathematical Reviews (MathSciNet): MR1264417
W. T. Gan, B. Gross and G. Savin, Fourier Coefficients of Modular Forms on $\mbox\upshape G_2$ , preprint (2001).
Mathematical Reviews (MathSciNet): MR1932327
Digital Object Identifier: doi:10.1215/S0012-7094-02-11514-2
Project Euclid: euclid.dmj/1085598120
Zentralblatt MATH: 1165.11315
G. B. Gurevich, Foundations of the Theory of Algebraic Invariants, translated by J. R. M. Radok and A. J. M. Spencer, P. Noordhoff, Groningen (1964).
Mathematical Reviews (MathSciNet): MR183733
Zentralblatt MATH: 0128.24601
A. Gyoja and Y. Omoda, Characteristic cycles of certain character sheaves , Indag. Math. (N.S.), 12 (2001), 329–335.
Mathematical Reviews (MathSciNet): MR1914084
Digital Object Identifier: doi:10.1016/S0019-3577(01)80014-8
Zentralblatt MATH: 1013.17017
A. C. Kable, Classes of integral $3$-tensors on $2$-space , Mathematika, 47 (2000), 205–217.
Mathematical Reviews (MathSciNet): MR1924498
A. C. Kable, The concomitants of a prehomogeneous vector space , to appear in J. Algebra.
Mathematical Reviews (MathSciNet): MR2022484
Digital Object Identifier: doi:10.1016/j.jalgebra.2003.02.004
Zentralblatt MATH: 1048.14033
A. C. Kable and A. Yukie, Prehomogeneous vector spaces and field extensions II , Invent. Math., 130 (1997), 315–344.
Mathematical Reviews (MathSciNet): MR1474160
Digital Object Identifier: doi:10.1007/s002220050187
Zentralblatt MATH: 0889.12004
A. C. Kable and A. Yukie, On the Number of Quintic Fields , preprint (2002).
Mathematical Reviews (MathSciNet): MR2138067
Digital Object Identifier: doi:10.1007/s00222-004-0391-2
Zentralblatt MATH: 1072.11082
N. Kawanaka, Generalized Gel'fand-Graev representations of exceptional simple algebraic groups over a finite field I , Invent. math., 84 (1986), 575–616.
Mathematical Reviews (MathSciNet): MR837529
Digital Object Identifier: doi:10.1007/BF01388748
T. Kimura, F. Sato and X.-W. Zhu, On the poles of $p$-adic complex powers and the $b$-functions of prehomogeneous vector spaces , Amer. J. Math., 112 (1990), 423–437.
Mathematical Reviews (MathSciNet): MR1055652
T. Kimura and M. Sato, A classification of irreducible prehomogeneous vector spaces and their relative invariants , Nagoya Math. J., 65 (1977), 1–155.
Mathematical Reviews (MathSciNet): MR430336
F. Knop and G. Menzel, Duale Varietäten von Fahnenvarietäten , Comment. Math. Helv., 62 (1987), 38–61.
Mathematical Reviews (MathSciNet): MR882964
G. Lusztig, Introduction to character sheaves , 165–179, The Arcata Conference on Representations of Finite Groups, Proc. Sympos. Pure Math., vol. 47, part 1, Amer. Math. Soc., Providence (1987).
Mathematical Reviews (MathSciNet): MR933358
Zentralblatt MATH: 0649.20038
M. Muro, M. Sato and T. Shintani, Theory of prehomogeneous vector spaces (algebraic part) – the English translation of Sato's lecture from Shintani's note , Nagoya Math. J., 120 (1990), 1–34.
Mathematical Reviews (MathSciNet): MR1086566
W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, PWN, Warsaw (1974).
Mathematical Reviews (MathSciNet): MR347767
Zentralblatt MATH: 0276.12002
I. Ozeki, On the microlocal structure of the regular prehomogeneous vector space associated with $\SL(5) \times \GL(4)$ I , Proc. Japan Acad. 55, Ser. A (1979), 37–40.
Mathematical Reviews (MathSciNet): MR528224
I. Ozeki, On the microlocal structure of the regular prehomogeneous vector space associated with $\SL(5) \times \GL(4)$ I , Publ. Res. Inst. Math. Sci., 26 , no. 3 (1990), 539–584.
Mathematical Reviews (MathSciNet): MR1068867
R. Scharlau, Paare alternierender Formen , Math. Z., 147 (1976), 13–19.
Mathematical Reviews (MathSciNet): MR419484
Digital Object Identifier: doi:10.1007/BF01214270
Zentralblatt MATH: 0304.15008
W. Schmidt, Number fields of given degree and bounded discriminant , Columbia University Number Theory Seminar (New York 1992), Astérisque, 228 (1995), 189–195.
Mathematical Reviews (MathSciNet): MR1330934
Waterloo Maple Inc., “Maple 7”, copyright 2001.
D. Witte, A. Yukie and R. Zierau, Prehomogeneous vector spaces and ergodic theory II , Trans. Amer. Math. Soc., 352 (2000), 1687–1708.
Mathematical Reviews (MathSciNet): MR1475697
Digital Object Identifier: doi:10.1090/S0002-9947-99-02224-2
Zentralblatt MATH: 0994.11040
D. Wright and A. Yukie, Prehomogeneous vector spaces and field extensions , Invent. math., 110 (1992), 283–314.
Mathematical Reviews (MathSciNet): MR1185585
Digital Object Identifier: doi:10.1007/BF01231334
Zentralblatt MATH: 0803.12004
A. Yukie, Density theorems related to prehomogeneous vector spaces , Automorphic forms, automorphic representations and automorphic L-functions over algebraic groups, Sūrikaisekikenkyūsho Kōkyūroku, 1173 (2000), 171–183.
Mathematical Reviews (MathSciNet): MR1840077
A. Yukie, Shintani Zeta Functions, Lond. Math. Soc. Lecture Notes, Vol. 183, Cambridge UP, Cambridge (1993).
Mathematical Reviews (MathSciNet): MR1267735
Zentralblatt MATH: 0801.11021