Glasnik Matematicki, Vol. 43, No.2 (2008), 293-307.

THE ADDITIVE UNIT STRUCTURE OF COMPLEX BIQUADRATIC FIELDS

Volker Ziegler

Graz University of Technology, Institute for analysis and computational number theory, Steyrergasse 30, A-8010 Graz, Austria
e-mail: ziegler@finanz.math.tugraz.at


Abstract.   We determine which rings of the form Z[α] are generated by their units, where α is a root of the polynomial X4 - BX2 + D such that α and all its conjugates are complex.

2000 Mathematics Subject Classification.   11R16, 11R27, 11A67.

Key words and phrases.   Biquadratic fields, unit sum number.


Full text (PDF) (free access)

DOI: 10.3336/gm.43.2.05


References:

  1. N. Ashrafi and P. Vámos, On the unit sum number of some rings, Q. J. Math. 56 (2005), 1-12.
    MathSciNet     CrossRef

  2. P. Belcher, Integers expressible as sums of distinct units, Bull. Lond. Math. Soc. 6 (1974), 66-68.
    MathSciNet     CrossRef

  3. P. Belcher, A test for integers being sums of distinct units applied to cubic fields, J. Lond. Math. Soc. (2) 12 (1976), 141-148.
    MathSciNet     CrossRef

  4. A. Filipin, R. Tichy, and V. Ziegler, The additive unit structure of purely quartic complex fields, Funct. Approx. Comment. Math., to appear.

  5. B. Goldsmith, S. Pabst, and A. Scott, Unit sum numbers of rings and modules, Q. J. Math. Oxford Ser. (2) 49 (1998), 331-344.
    MathSciNet     CrossRef

  6. B. Jacobson, Sums of distinct divisors and sums of distinct units, Proc. Am. Math. Soc. 15 (1964), 179-183.
    MathSciNet     CrossRef

  7. M. Jarden and W. Narkiewicz, On sums of units, Monatsh. Math. 150 (2007), 327-332.
    MathSciNet     CrossRef

  8. J. Sliwa, Sums of distinct units, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 22 (1974), 11-13.
    MathSciNet

  9. R. Tichy and V. Ziegler, Units generating the ring of integers of complex cubic fields, Colloq. Math. 109 (2007), 71-83.
    MathSciNet

  10. P. Vámos, 2-good rings, Q. J. Math. 56 (2005), 417-430.
    MathSciNet     CrossRef

  11. L. C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics 83, Springer-Verlag, New York, 1997.
    MathSciNet

  12. D. Zelinsky, Every linear transformation is a sum of nonsingular ones, Proc. Am. Math. Soc. 5 (1954), 627-630.
    MathSciNet     CrossRef


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