Ideal Class Groups of Iwasawa‐Theoretical Abelian Extensions Over the Rational Field

Throughout this paper, we shall suppose that all algebraic number fields, namely, all algebraic extensions over the rational field Q, are contained in the complex field C. Let P be the set of all prime numbers. For any algebraic number field F , let C F denote the ideal class group of F and, writing F + for the maximal real subfield of F , letC F −denote the kernel of the norm map from C F to the ideal class group of F + ; for each l ∈ P , let C F ( l ) denote the l ‐class group of F , that is, the l ‐primary component of C F , and letC F − ( l )denote the l ‐primary component ofC F − . Furthermore, for each l ∈ P , we denote by Z l the ring of l ‐adic integers.