Totally positive units and squares
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- by I. Hughes and R. Mollin PDF
- Proc. Amer. Math. Soc. 87 (1983), 613-616 Request permission
Abstract:
Let $K$ be a finite cyclic extension of the rational number field $Q$, with Galois group $G(K/Q)$ of order ${p^a}$ for an odd prime $p$. Armitage and Fröhlich [1] proved that if the order of 2 modulo $p$ is even and the class number ${h_K}$ of $K$ is odd then $U_K^ + = U_K^2$, where ${U_K}$ is the group of units of the ring of integers ${\mathcal {C}_K}$ of $K$, $U_K^ +$ is the group of totally positive units, and $U_K^2$ is the group of unit squares. The purpose of this paper is to provide a generalization of this result to a larger class of abelian extensions of ${Q.^2}$References
-
J. V. Armitage and A. Fröhlich, Class numbers and unit signatures, Mathematica 14 (1967), 94-98.
- Dennis A. Garbanati, Unit signatures, and even class numbers, and relative class numbers, J. Reine Angew. Math. 274(275) (1975), 376–384. MR 376612, DOI 10.1515/crll.1975.274-275.376
- Georges Gras, Critère de parité du nombre de classes des extensions abéliennes réelles de $Q$ de degré impair, Bull. Soc. Math. France 103 (1975), no. 2, 177–190 (French). MR 387238
- Helmut Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (German). MR 0049239 E. Hecke, Vorlesungen über die Theorie der algebaischen Zahlen, Chelsea, New York, 1948.
- Kenkichi Iwasawa, A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg 20 (1956), 257–258. MR 83013, DOI 10.1007/BF03374563
- Gerald J. Janusz, Algebraic number fields, Pure and Applied Mathematics, Vol. 55, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1973. MR 0366864
- H. W. Leopoldt, Über Einheitengruppe und Klassenzahl reeller abelscher Zahlkörper, Abh. Deutsch. Akad. Wiss. Berlin. Kl. Math. Nat. 1953 (1953), no. 2, 48 pp. (1954) (German). MR 0067927
- R. A. Mollin, Class numbers and a generalized Fermat theorem, J. Number Theory 16 (1983), no. 3, 420–429. MR 707613, DOI 10.1016/0022-314X(83)90068-9 B. Oriat, Sur l’article de Leopoldt intitulé (über Einheitengruppe und Klassenzahl reeller abelscher Zahlkorper), Publ. Math. de la Faculté des Sciences de Besancon, Théorie des nombres (1974-75). H. Weber, Lehubuch der Algebra. Vol. II, Aufl. Braunschweig, 1899; reprinted by Chelsea, New York, 1966.
Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 87 (1983), 613-616
- MSC: Primary 12A45; Secondary 12A35, 12A95
- DOI: https://doi.org/10.1090/S0002-9939-1983-0687627-7
- MathSciNet review: 687627