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On a Problem of Hasse for Certain Imaginary Abelian Fields

✍ Scribed by Yasuo Motoda; Toru Nakahara; Syed Inayat Ali Shah

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
125 KB
Volume
96
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let K be the composite field of an imaginary quadratic field QðoÞ of conductor d and a real abelian field L of conductor f distinct from the rationals Q; where ðd; f Þ ¼ 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 Besanc¸on, Theor. Nombres (1984) 25pp) and with monogenic (3).

(1) If QðoÞathe Gau field QðiÞ; then Z K is of non-monogenesis.

(2) If QðoÞ ¼ QðiÞ; then for a sextic field K; Z K is of non-monogenesis except for two fields K of conductors 28 and 36.

(3) Let QðoÞ ¼ QðiÞ: If Z K has a power basis, then Z L must have a power basis. Conversely, let L be the maximal real subfield k þ 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. # 2002 Elsevier Science (USA)

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