Study of Cyclotomic Iwasawa Theory.

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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.