Class Number Problem for Imaginary Cyclic Number Fields

Abstract

LetNbe an imaginary cyclic number field of degree 2n. Whenn=3 orn=2m⩾2, the fieldsNwith class numbers equal to their genus class numbers and the fieldsNwith relative class numbers less than or equal to 4 are completely determined [10, 13, 26, 27]. Now assume thatn⩾5 andnis not a 2-power. In this paper, first we determine all imaginary cyclic number fields of degree 2nwith relative class numbers less than or equal to 4. Second, we determine all imaginary cyclic number fields of degree 2nwith class numbers equal to their genus class numbers.