On the Birch-Tate conjecture for cyclic number fields

Abstract

The proof of the Main Conjecture in Iwasawa theory by Mazur and Wiles implies that the Birch-Tate conjecture # K 2( O F ) = w 2( F)| ζ F (−1)| is true up to 2-torsion for real abelian number fields. In the present paper the even part of the Birch-Tate conjecture is verified for cyclic number fields F with odd prime degree p if the order of 2 mod p is even, 2 is inertial in F and the class number h( F) of F is odd.