Determination of all nonquadratic imaginary cyclic number fields of 2-power degrees with ideal class groups of exponents ≤2

Abstract

We determine all nonquadratic imaginary cyclic number fieldsKof 2-power degrees with ideal class groups of exponents≤2\leq 2, i.e., with ideal class groups such that the square of each ideal class is the principal class, i.e., such that the ideal class groups are isomorphic to some(Z/2Z)m,m≥0{({\mathbf {Z}}/2{\mathbf {Z}})^m},m \geq 0. There are 38 such number fields: 33 of them are quartic ones (see Theorem 13), 4 of them are octic ones (see Theorem 12), and 1 of them has degree 16 (see Theorem 11).