On Capitulation of S-Ideals in Zp-Extensions

✍ Scribed by Hiroki Sumida

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 167 KB
• Volume: 86
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.2000.2561

✦ Synopsis

Let k be a finite extension of Q and p a prime number. Let K be a Z p -extension of k and S the set of all prime ideals in k which are ramified in K. We denote by A$ the p-Sylow subgroup of the S-divisor class group of K. We give a criterion for A$ =0 which can be applied for general Z p -extensions. Further, we especially investigate the criterion for a totally real number field k in which p splits completely.

2001 Academic Press

1. Introduction

Let k be a finite extension of Q and p a prime number. Let K be a Z p -extension of k and k n /K the unique cyclic extension of k of degree p n . Further, let S be the set of all prime ideals in k which are ramified in K. By Theorem 1 in [11], all prime ideals in S lie above p. We assume that all prime ideals in S are fully ramified in K. We denote by A n the p-Sylow subgroup of the ideal class group of k n . We put A = A n , where the map: A n Ä A m is induced by the natural inclusion map i n, m : k n / Ä k m for m n. We will denote the induced maps by i n, m . Similarly, we denote by A$ n the p-Sylow subgroup of the S-ideal class group of k n and put