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We determine all nonquadratic imaginary cyclic number fields  K  of 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 / 2 Z ) 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).

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