Class Number Problem for Imaginary Cyclic Number Fields

✍ Scribed by Ku-Young Chang; Soun-Hi Kwon

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 292 KB
• Volume: 73
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1998.2298

✦ Synopsis

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 this paper, first we determine all imaginary cyclic number fields of degree 2n with relative class numbers less than or equal to 4. Second, we determine all imaginary cyclic number fields of degree 2n with class numbers equal to their genus class numbers.

1998 Academic Press

1. Introduction

Let N be an imaginary abelian number field and let N + be its maximal real subfield. We know that the class number h N of N is divisible by h N + , that of N + . The ratio h N Âh N + is called the relative class number of N, which is denoted by h & N . It is known that h & N goes to the infinity as f N , the conductor N, approaches to the infinity. Thus there exist only finitely many imaginary abelian number fields with given relative class number. Throughout this paper l is an odd prime number and n is an integer such that n 5 and n is not a 2-power. From now on we shall consider only the imaginary cyclic number fields of degree 2n. The first goal of this paper is to determine all imaginary cyclic number fields with relative class numbers less than or equal to 4: