Local units of integer group ring of cyclic group of order 64 for character with character field ${\mathbb Q}_{64}$

R. Zh. Aleev ; O. V. Mitina ; T. A. Khanenko

Čelâbinskij fiziko-matematičeskij žurnal, Tome 3 (2018) no. 3, pp. 253-275

Résumé

The work is devoted to the study of units of the integer group ring for order $64$ cyclic group. The units groups of the integer group rings for the cyclic groups of the orders $2$ and $4$ are trivial, for the order $8$ this group is well known, for the cyclic group of the order $16$ such group is described earlier. The study of units of the integer group ring of the order $64$ cyclic group is carried out in terms of local units defined by the characters of the order 64 cyclic group and by units of the ring of the circular field $ {\mathbb Q}_{64}$, obtained by adjoining the degree 64 primitive root of $1$ to the field of the rational numbers. The most important role among the local units is played by units for the character with the character field ${\mathbb Q}_{64}$, because they provide the possibility of the inductive approach to the description of the units groups of the integer group rings for the cyclic $2$-groups. We note that earlier, by direct calculations, the authors obtained a description of the local units for a character with the character field ${\mathbb Q}_{32}$ of the integer group ring for the cyclic group of the order $32$. Therefore, the next natural step is to study the local units for a character with the character field ${\mathbb Q}_{64}$ of the integer group ring for the order 64 cyclic group. To achieve these goals, a new approach has been developed, which can be applied to units groups of the integer group rings for the cyclic $2$-groups of an order greater than $64$.

Détail

Keywords: group ring, unit of group ring, cyclic group, cyclotomic field, integer group ring.