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The class-number one problem for some real cubic number fields with negative discriminants

✍ Scribed by Stéphane R. Louboutin

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
136 KB
Volume
121
Category
Article
ISSN
0022-314X

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✦ Synopsis

We prove that there are effectively only finitely many real cubic number fields of a given class number with negative discriminants and ring of algebraic integers generated by an algebraic unit. As an example, we then determine all these cubic number fields of class number one. There are 42 of them. As a byproduct of our approach, we obtain a new proof of Nagell's result according to which a real cubic unit > 1 of negative discriminant is generally the fundamental unit of the cubic order Z[ ].

📜 SIMILAR VOLUMES

Class Number One Problem for Imaginary F
Class Number One Problem for Imaginary Function Fields: The Cyclic Prime Power Case
✍ Stéphan Sémirat 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 171 KB

In this paper, we determine all finite separable imaginary extensions KÂF q (x) whose maximal order is a principal ideal domain in case KÂF q (x) is a non zero genus cyclic extension of prime power degree. There exist exactly 42 such extensions, among which 7 are non isomorphic over F q . 2000 Acad