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Class Groups , Totally Positive Units , and Squares
| Content Provider | Semantic Scholar |
|---|---|
| Author | Edgar, H. Gillock Ii Mollin, Richard A. Peterson, Bercedis L. |
| Copyright Year | 1986 |
| Abstract | Given a totally real algebraic number field K, we investigate when totally positive units, U¿, are squares, u£. In particular, we prove that the rank of U¿ /Ují is bounded above by the minimum of (1) the 2-rank of the narrow class group of K and (2) the rank of Ul /U¿ as L ranges over all (finite) totally real extension fields of K. Several applications are also provided. 1. Notation and preliminaries. Let K be an algebraic number field and let CK denote the ideal class group in the ordinary or "wide" sense. Let CK+) denote the "narrow" ideal class group of A". Thus \CK\ = hK, the "wide" class number of K, and \CK+)\ = h(K+), the "narrow" class number of K. We denote the Hubert class field of K by A"(1); i.e., Gal(A"(1)/A") s CK, and we denote the "narrow" Hilbert class field by A~(+); i.e., Gal(A~(+)/A") s CK+\ Moreover we adopt the "bar" convention to mean "modulo squares"; for example, CK = CK/C\. Let UK denote the group of units of the ring of algebraic integers of K. When K is totally real, we let Ux denote the subgroup of totally positive units; i.e., those units u such that ua > 0 for all embeddings a of A' into R. Finally, for any finite abelian group A with \A\ = 2d, d is called the 2-rank of A, which we denote by dim2 A. 2. Results. We are concerned with the question: (*) When is U¿ = U21 We begin by observing that dim2(¿7¿) = 0 if and only if A"( + )=A"(1) [6, Theorem 3.1, p. 203]. In particular, when A" is a real finite Galois extension of 2-power degree over Q, then dim2(Ux) = 0 if and only if N(UK) = (±1) [3, Theorem 1, p. 166]. For example, when A is a real quadratic field, then dim2(U¿) = 0 if and only if the norm of the fundamental unit is -1. Necessary and sufficient conditions (in terms of the arithmetic of the underlying quadratic field K ) for the existence of a fundamental unit of norm -1 are unknown (see [8]). This indicates the difficulty of solving (*) for the simplest even degree case. In this regard one may ask whether (*) is equivalent to such a norm statement for other fields. In a recent letter to the authors, V. Ennola answered (*) for cyclic cubic fields K as follows: Let e be a norm positive unit of A" such that -1 and the conjugates of e generate the unit group. Then dim2(Ux) = 0 if and only if e is not totally positive. However, as with Received by the editors February 18, 1985, and, in revised form, September 20, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 11R80, 11R27, 11R29; Secondary 11R37,11R32. 1 This author's research is supported by N.S.E.R.C. Canada. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://www.ams.org/journals/proc/1986-098-01/S0002-9939-1986-0848870-X/S0002-9939-1986-0848870-X.pdf |
| Alternate Webpage(s) | http://people.ucalgary.ca/~ramollin/cgtpus.pdf |
| Language | English |
| Access Restriction | Open |
| Subject Keyword | Belief revision Cubic function Degree (graph theory) Immunostimulating conjugate (antigen) Linear algebra Mathematics Subject Classification Modulo operation Neoplasm Metastasis Revision procedure SNORD21 gene Subgroup A Nepoviruses Unit |
| Content Type | Text |
| Resource Type | Article |
