On the ideal class groups of imaginary abelian fields with small conductor

Abstract

Let k k be any imaginary abelian field with conductor not exceeding 100, where an abelian field means a finite abelian extension over the rational field Q {\mathbf {Q}} contained in the complex field. Let C ( k ) C(k) denote the ideal class group of k k , C − ( k ) {C^ - }(k) the kernel of the norm map from C ( k ) C(k) to the ideal class group of the maximal real subfield of k k , and f ( k ) f(k) the conductor of k ; f ( k ) ⩽ 100 k;f(k) \leqslant 100 . Proving a preliminary result on 2 2 -ranks of ideal class groups of certain imaginary abelian fields, this paper determines the structure of the abelian group C − ( k ) {C^ - }(k) and, under the condition that either [ k : Q ] ⩽ 23 [k:{\mathbf {Q}}] \leqslant 23 or f ( k ) f(k) is not a prime ⩾ 71 \geqslant 71 , determines the structure of C ( k ) C(k) .