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Fundamental units of certain algebraic number fields

Fundamental units of certain algebraic number fields By MAxoTo ISHIDA 0. We consider an irreducible cubic equation (1) /(x) = x8 + kx" + 1 x -- 1 = o (t:, l ~ Z) with negative discriminant D I < 0. Let ~ be a (unique) real root of /(X). Then K ---- Q (~) is a (real but not totally-real) cubic number field and ~ is a unit of K. Moreover let L be the Galois closure of K, i. e. the smallest splitting field of the equation (1). Clearly L is a Galois extension, over Q, of degree 6, whose Galois group G ---- Gal (L/Q) is isomorphic to ~3 (the symmetric group of three letters). In this paper, we investigate fundamental units of K and L under some assumptions on/c and 1. Note that the Dirichlet numbers of K and L are 1 and 2 respectively. Let K0 = Q (/~/) be a (unique) quadratic subfield of L. 1. Throughout this section, we assume that the following conditions are satisfied: (a) /c ~-- l -- 0 (rood 2), and --D~/3:4= square in g. (b) K0 ~ Q(VZ3) i. e. Remark 1. Of course, the conditions (a) and (b) are easily checked when the http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg Springer Journals

Fundamental units of certain algebraic number fields

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general; Algebra; Differential Geometry; Number Theory; Topology; Geometry
ISSN
0025-5858
eISSN
1865-8784
DOI
10.1007/BF02992833
Publisher site
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Abstract

By MAxoTo ISHIDA 0. We consider an irreducible cubic equation (1) /(x) = x8 + kx" + 1 x -- 1 = o (t:, l ~ Z) with negative discriminant D I < 0. Let ~ be a (unique) real root of /(X). Then K ---- Q (~) is a (real but not totally-real) cubic number field and ~ is a unit of K. Moreover let L be the Galois closure of K, i. e. the smallest splitting field of the equation (1). Clearly L is a Galois extension, over Q, of degree 6, whose Galois group G ---- Gal (L/Q) is isomorphic to ~3 (the symmetric group of three letters). In this paper, we investigate fundamental units of K and L under some assumptions on/c and 1. Note that the Dirichlet numbers of K and L are 1 and 2 respectively. Let K0 = Q (/~/) be a (unique) quadratic subfield of L. 1. Throughout this section, we assume that the following conditions are satisfied: (a) /c ~-- l -- 0 (rood 2), and --D~/3:4= square in g. (b) K0 ~ Q(VZ3) i. e. Remark 1. Of course, the conditions (a) and (b) are easily checked when the

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Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg – Springer Journals

Published: Nov 17, 2008

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