Humio Ichimura

On the ring of integers of a tame Kummer extension over a number field

Abstract Gomez Ayala gave a necessary and sufficient condition for a tame Kummer extension of prime degree over a number field to have a relative normal integral basis (NIB for short). We generalize this result for a tame cyclic Kummer extension of arbitrary degree, and then prove the following “capitulation” theorem for the Galois module structure of rings of integers. Let m⩾2 be an integer, and F a number field. Then, there exists a finite extension L/F depending on m and F such that for any abelian extension K/F of exponent dividing m, the pushed-up extension LK/L has a NIB.

On the parity of the class number of the 7nth cyclotomic field

AbstractFor an odd prime number p and an integer n ≥ 0, let hn be the class number of the pn+1st cyclotomic field Q(

Note on the Class Number of the pth Cyclotomic Field, II

\zeta _{p^{n + 1} }

A note on quadratic fields in which a fixed prime number splits completely

). It is known that when p = 3 or 5, hn is odd for all n ≥ 0. We prove that the same holds also when p = 7.

ON THE 2-ADIC IWASAWA LAMBDA INVARIANTS OF THE p-CYCLOTOMIC FIELDS AND THEIR QUADRATIC TWISTS

AbstractWe prove that the class number of the real quadratic field

On some congruences for units of localp-cyclotomic fields

{\mathbb{Q}}\left( {\sqrt {a^{2n} + 4} } \right)