Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665812016On a duality of Gras between totally positive and primary cyclotomic units125132ENHumioIchimuraLet K be a real abelian field of odd degree over Q, and C the group of cyclotomic units of K. We denote by C+ and C0 the totally positive and primary elements of C, respectively. G. Gras found a duality between the Galois modules C+/C2 and C0/C2 by some ingenious calculation on cyclotomic units. We give an alternative proof using a consequence (=“Gras conjecture”) of the Iwasawa main conjecture and the standard reflection argument. We also give some related topics.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15665412012HILBERT-SPEISER NUMBER FIELDS AND STICKELBERGER IDEALS; THE CASE p = 23348ENHumioIchimuraWe say that a number field F satisfies the condition (H′<sub>2<sup>m</sup></sub>) when any abelian extension of exponent dividing 2<sup>m </sup> has a normal basis with respect to rings of 2-integers. We say that it satisfies (H′
<sub>2<sup>∞</sup></sub>) when it satisfies (H′
<sub>2<sup>m</sup></sub>) for all m. We give a condition for F to satisfy (H'<sub>2<sup>m</sup></sub>), and show that the imaginary quadratic fields F = Q(√−1) and Q(√−2) satisfy the very strong condition (H′
<sub>2<sup>∞</sup></sub>) if the conjecture that h<sup>+</sup><sub>2<sup>m</sup></sub> = 1 for all m is valid. Here, h<sup>+</sup><sub>2<sup>m</sup></sub>) is the class number of the maximal real abelian field of conductor 2<sup>m</sup>.No potential conflict of interest relevant to this article was reported.Department of Mathematics, Faculty of Science, Okayama UniversityActa Medica Okayama0030-15664812006Stickelberger Ideals and Normal Bases of Rings of p-integersENHumioIchimuraNo potential conflict of interest relevant to this article was reported.