Capitulation in unramified quadratic extensions of real quadratic number fields

Abstract

Letkbe an algebraic number field andCkits ideal class group (in the wider sense). SupposeKis a finite extension ofk. Then we say that an ideal class ofk capitulatesinKif this class is in the kernel of the homomorphisminduced by extension of ideals fromktoK(See Section 2 below). In [4], Iwasawa gives examples of real quadratic number fields,with distinct primesPi≡ 1 (mod 4), for which all the ideal classes of the 2-class group,Ck,2(the 2-Sylow subgroup ofCk), capitulate in an unramified quadratic extension ofk. In these examples,Ck,2is abelian of type (2,2), i.e. isomorphic to ℤ/2ℤ×ℤ/2ℤ and so all four ideal classes capitulate.