Uniform distribution of the Steinitz invariants of quadratic and cubic extensions

Compositio Mathematica

Volumn 142, Issue 1, 2006, Pages 84-100

  Kable, Anthony C ^a   Wright, David J ^a  

^a OKLAHOMA STATE UNIVERSITY   (United States)

Author keywords

Cubic extension; Equidistribution; Shintani zeta function; Steinitz invariant

Indexed keywords

EID: 33845302621     PISSN: 0010437X     EISSN: None     Source Type: Journal    
DOI: 10.1112/S0010437X05001740     Document Type: Article

References (25)

1. Art65 E. Artin, Questions de base minimale dans la théorie des nombres algébriques, in Collected papers of Emil Artin (Addison-Wesley, Reading, MA, 1965), 229-321.
2. BBP04 K. Belabas, M. Bhargava and C. Pomerance, Error estimates for the Davenport-Heilbronn theorems, Preprint (2004).
3. Bou65 N. Bourbaki, Algèbre Commutative, Actualités Scientifiques et Industrielles, vol. 1314 (Hermann, Paris, 1965).
4. Car98 J. E. Carter, Module structure of integers in metacyclic extensions, Colloq. Math. 76 (1998), 191-199.
5. Coh00 H. Cohen, Advanced Topics in Computational Number Theory, Graduate Texts in Mathematics, vol. 193 (Springer, New York, 2000).
6. DW86 B. Datskovsky and D. J. Wright, The adelic zeta function associated to the space of binary cubic forms II: Local theory, J. reine angew. Math. 367 (1986), 27-75.
7. DW88 B. Datskovsky and D. J. Wright, Density of discriminants of cubic extensions, J. reine angew. Math. 386 (1988), 116-138.
8. DF64 B.N. Delone and D.K. Fadeev, Theory of Irrationalities of the Third Degree, Translated Mathematical Monographs, vol. 10 (American Mathematical Society, Providence, RI, 1964).
9. Fro60 A. Fröhlich, The discriminants of relative extensions and the existence of integral bases, Mathematika 7 (1960), 15-22.
10. FT91 A. Fröhlich and M. J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics, vol. 27 (Cambridge University Press, Cambridge, 1991).
11. God02 M. Godin, Structure Galoisienne d'anneaux d'entiers, Thesis, L'Université de Valenciennes et du Hainaut Cambrésis, 2002.
12. GS03 M. Godin and B. Sodaïgui, Module structure of rings of integers in octahedral extensions, Acta Arith. 109 (2003), 321-327.
13. KP00 C. Khare and D. Prasad, On the Steinitz module and capitulation of ideals, Nagoya Math. J. 160 (2000), 1-15.
14. Lan86 S. Lang, Algebraic Number Theory, Graduate Texts in Mathematics, vol. 110 (Springer, New York, 1986).
15. Rob01 D. P. Roberts, Density of cubic field discriminants, Math. Comp. 70 (2001), 1699-1705.
16. SS74 M. Sato and T. Shintani, On zeta functions associated with prehomogeneous vector spaces, Ann. of Math. (2) 100 (1974), 131-170.
17. Shi72 T. Shintani, On Dirichlet series whose coefficients are class-numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132-188.
18. Sov02 E. Soverchia, Steinitz classes of metacyclic extensions, J. London Math. Soc. (2) 66 (2002), 61-72.
19. Wei74 A. Weil, Basic Number Theory (Springer, Berlin, 1974).
20. Wri85 D. J. Wright, The adelic zeta function associated with the space of binary cubic forms I: Global theory, Math. Ann. 270 (1985), 503-534.
21. Wri89 D. J. Wright, Distribution of discriminants of abelian extensions, Proc. London Math. Soc. (3) 58 (1989), 17-50.
22. WY92 D. J. Wright and A. Yukie, Prehomogeneous vector spaces and field extensions, Invent. Math. 110 (1992), 283-314.
23. Yuk93 A. Yukie, Shintani zeta functions, London Mathematical Society Lecture Notes, vol. 183 (Cambridge University Press, Cambridge, 1993).
24. Yuk00 A. Yukie, Rational orbit decompositions of prehomogeneous vector spaces, Tohoku University Lecture Notes, June 2000, available at http://www.math.tohoku.ac.jp/~yukie/.
25. Yuk03 A. Yukie, On the density of cubic fields, Preprint (2003).