Gaussian periods and units in certain cyclic fields

Lazarus, Andrew J.

doi:10.1090/S0002-9939-1992-1093600-5

Gaussian periods and units in certain cyclic fields
HTML articles powered by AMS MathViewer

by Andrew J. Lazarus

Proc. Amer. Math. Soc. 115 (1992), 961-968

DOI: https://doi.org/10.1090/S0002-9939-1992-1093600-5

PDF | Request permission

Abstract:

We analyze the property of period-unit integer translation (there exists a Gaussian period $\eta$ and rational integer $c$ such that $\eta + c$ is a unit) in simplest quadratic, cubic, and quartic fields of arbitrary conductor. This is an extension of work of E. Lehmer, R. Schoof, and L. C. Washington for prime conductor. We also determine the Gaussian period polynomial for arbitrary conductor.

References

Die Lehre von der Kreisteilung

Albert Châtelet, Arithmétique des corps abéliens du troisième degré, Ann. Sci. École Norm. Sup. (3) 63 (1946), 109–160 (1947) (French). MR 0020598

Theoria residuorum biquadraticorum: commentatio prima

Table numérique du nombre de classes et des unités des extensions cycliques de degré

de

Kenneth Hardy, R. H. Hudson, D. Richman, Kenneth S. Williams, and N. M. Holtz, Calculation of the class numbers of imaginary cyclic quartic fields, Math. Comp. 49 (1987), no. 180, 615–620. MR 906194, DOI 10.1090/S0025-5718-1987-0906194-5

Arithmetische Bestimmung von Grundeinheit und Klassenzahl in zyklischen kubischen und biquadratischen Zahlkörpern

Kenneth F. Ireland and Michael I. Rosen, A classical introduction to modern number theory, Graduate Texts in Mathematics, vol. 84, Springer-Verlag, New York-Berlin, 1982. Revised edition of Elements of number theory. MR 661047
Andrew J. Lazarus, Class numbers of simplest quartic fields, Number theory (Banff, AB, 1988) de Gruyter, Berlin, 1990, pp. 313–323. MR 1106670
Andrew J. Lazarus, On the class number and unit index of simplest quartic fields, Nagoya Math. J. 121 (1991), 1–13. MR 1096465, DOI 10.1017/S0027763000003378
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
Toru Nakahara, On cyclic biquadratic fields related to a problem of Hasse, Monatsh. Math. 94 (1982), no. 2, 125–132. MR 678047, DOI 10.1007/BF01301930
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
Daniel Shanks, The simplest cubic fields, Math. Comp. 28 (1974), 1137–1152. MR 352049, DOI 10.1090/S0025-5718-1974-0352049-8