On a problem of Hasse for certain imaginary Abelian fields

JOURNAL OF NUMBER THEORY

卷:96

Article

Motoda, Y ; Nakahara, T ; Shah, SIA

关键词: Hasse's problem;    non-monogenesis;    power basis;    discriminant;    abelian field;    imaginary quadratic field;   

DOI  :  10.1006/jnth.2002.2805

来源: Elsevier

【 摘 要 】

Let K be the composite field of an imaginary quadratic field Q(omega) of conductor d and a real abelian field L of conductor f distinct from the rationals Q, where (d,f) = 1. Let Z(K) be the ring of integers in K. Then concerning to Hasse's problem we construct new families of infinitely many fields K with the non-monogenic phenomena (1), (2) which supplement (J. Number Theory 23 (1986), 347-353; Publ. Math. Fac. Sci Besan on, Theor. Nombres (1984) 25pp) and with monogenic (3). (1) If Q(omega) not equal the Gauss field Q(i), then Z(K) is of non-monogenesis. (2) If Q(omega) = Q(i), then for a sextic field K, Z(K) is of non-monogenesis except for two fields K of conductors 28 and 36. (3) Let Q(omega) = Q(i). If Z(K) has a power basis, then Z(L) must have a power basis. Conversely, let L be the maximal real subfield k(f)(+) of a cyclotomic field k(f), namely K be the maximal imaginary subfield of k(4f) of conductor 4f. Then Z(K) has a power basis. (C) 2002 Elsevier Science (USA).

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