✦ LIBER ✦

Class number one problem for normal CM-fields

✍ Scribed by Sun-Mi Park; Soun-Hi Kwon

Publisher: Elsevier Science
Year: 2007
Tongue: English
Weight: 360 KB
Volume: 125
Category: Article
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2006.10.012

No coin nor oath required. For personal study only.

✦ Synopsis

It is known that if we assume the Generalized Riemann Hypothesis, then any normal CM-field with relative class number one is of degree less than or equal to 96. All normal CM-fields of degree less than 48 with class number one are known. In addition, for normal CM-fields of degree 48 the class number one problem is partially solved. In this paper we will show that under the Generalized Riemann Hypothesis there is no more normal CM-fields with class number one except for the possible fields of degrees 64 or 96.

📜 SIMILAR VOLUMES

Class Number Problem for Imaginary Cycli

Class Number Problem for Imaginary Cyclic Number Fields

✍ Ku-Young Chang; Soun-Hi Kwon 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 292 KB

Let N be an imaginary cyclic number field of degree 2n. When n=3 or n=2 m 2, the fields N with class numbers equal to their genus class numbers and the fields N with relative class numbers less than or equal to 4 are completely determined [10,13,26,27]. Now assume that n 5 and n is not a 2-power. In

The class-number one problem for some re

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

✍ Stéphane R. Louboutin 📂 Article 📅 2006 🏛 Elsevier Science 🌐 English ⚖ 136 KB

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.

The Non-normal Quartic CM-Fields and the

The Non-normal Quartic CM-Fields and the Octic Dihedral CM-Fields with Relative Class Number Two

✍ Hee-Sun Yang; Soun-Hi Kwon 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 157 KB

In this paper we determine all non-normal quartic CM-fields with relative class number two and all octic dihedral CM-fields with relative class number two: there are exactly 254 non-isomorphic non-normal quartic CM-fields with relative class number two and 95 non-isomorphic octic dihedral CM-fields

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

Solution of the class number two problem

Solution of the class number two problem for cyclotomic fields

✍ John Myron Masley 📂 Article 📅 1975 🏛 Springer-Verlag 🌐 English ⚖ 117 KB

Cyclotomic Function Fields with Ideal Cl

Cyclotomic Function Fields with Ideal Class Number One

✍ Stéphan Sémirat 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 139 KB

We list all imaginary cyclotomic extensions ‫ކ‬ x, ⌳ r‫ކ‬ x with ideal class q M Ž x . q number equal to one. Apart from the zero genus ones, there are 17 solutions up to Ž . ‫ކ‬ x -isomorphism: 13 of them are defined over ‫ކ‬ and the 4 remainings are q 3 defined over ‫ކ‬ .