| Co-Investigator(Kenkyū-buntansha) |
SUMIDA Hiroki Hiroshima Univ., Faculty of Integrated Arts and Science, 総合科学部, 助手 (90291476)
KOYA Yoshihiro Yokohama City Univ., Graduate School of Integrated Science, Assoc.Prof., 総合理学研究科, 助教授 (50254230)
KAGAWA Hirotada Kagawa Univ., Faculty of Education, Prof., 教育学部, 教授 (00180224)
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| Research Abstract |
During 2001-2003, I studied, as follows, cyclotomic Iwasawa theory, normal integral basis problem and some relation between them. In what follows, p denotes a prime number.
(A)Let K be a real abelian field, K_∞/K the cyclotomic Z_p-extension, and K_n its n-th layer. Let A_n be the Sylow p-subgroup of the ideal class group of K_n, and A_∞ the natural injective limit of A_n. Let A_0 be the image of A_0 in A_∞. I proved that the capitulation cokernel A_∞/A_0 is isomorphic to a certain Galois group associated to K_∞, and gave a condition for A_∞/A_0 ={0}.
(B)Let K be an imaginary abelian field with ζ_p ∈ K, and K_∞ K_n be as in (A). Let a be a square free integer of the maximal real subfield K^+_n. I described for what m greater than or equal n, the cyclic extension K_m(a^<1/p>)/K_m has a relative normal integral basis (NIB) in terms of the p-adic L-function associated to K.
(C)Let F be a number field with ζ_p 【not a member of】 F, and K = F(ζ_p). I proved that an unramified cyclic extension N/F of degree p has a NIB if and only in NK/K has a NIB. When p = 3 and F is an imaginary quadratic field, this is already known by Brinkhuis. I gave some applications of this result.
(D)Gomez Ayala gave a necessary and sufficient condition for a Kummer extension of prime degree to have a NIB. I generalised this criterion for a general cyclic Kummer extension. As an application, I Showed a "capitulation" theorem for rings of integers of abelian extensions over a number field.
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