TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 311. Number I. January 1989 DETERMINATION OF ALL IMAGINARY CYCLIC QUARTIC FIELDS WITH CLASS NUMBER 2 KENNETH HARDY, RICHARD H. HUDSON, DAVID RICHMAN AND KENNETH S. WILLIAMS ABSTRACT. It is proved that there are exactly 8 imaginary cyclic quartic fields with class number 2. 1. INTRODUCTION Let K be an imaginary cyclic quartic extension of the rational field Q. K has a unique quadratic subfield which we denote by k. The class number of K (resp. k) is denoted by h(K) (resp. h(k)). The relative class number of Kover k is the positive integer h*(K) = h(K)/h(k). The conductor of K is denoted by f. In 1972 Uchida [17] proved that if K is a field with h * (K) = 1 then f < 50,000. In 1980 Setzer [14] computed the values of h*(K) for all fields K having h*(K) == 1 (mod2) for which f < 50,000. He found that h * (K) = 1 for exactly 7 fields K. Since h (k) = 1 for these 7 fields, Setzer's work completed the proof of the following theorem. Theorem 1 (Uchida-Setzer). If K is an imaginary cyclic quartic field of class number 1, then K is one of the 7 fields Q(J-(5+2v's)) Q (J-(13 + 2m)) Q (J-(2+V2)) (/ = 5), (/ = 13), (/=16), Received by the editors June 8, 1987 and, in revised form, January 25, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 12A50, 12A30; Secondary 12A35. Key words and phrases. Imaginary cyclic quartic fields, class number. Research of the first author was supported by Natural Sciences and Engineering Research Council of Canada Grant A-7823. Research of the fourth author was supported by Natural Sciences and Engineering Research Council of Canada Grant A-7233. © 1989 American Mathematical Society 0002·9947/89 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use