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.

Bibliographie

[1] Aleev R.Zh., Mitina O.V., Khristenko E.A., “Congruence modulo 2 of circular units in the fields $Q_{16}$ and $Q_{32}$”, Chelyabinsk Physical and Mathematical Journal, 1:4 (2016), 8–29 (In Russ.) | MR

[2] Aleev R.Zh., Mitina O.V., Khanenko T.A., “Finding of units for integer group rings of orders $16$ and $32$ cyclic groups”, Chelyabinsk Physical and Mathematical Journal, 1:4 (2016), 30–55 (In Russ.) | MR

[3] Aleev R.Zh., Mitina O.V., Khanenko T.A., “Description of the unit group of the integer group ring of a cyclic group of order $16$”, Proceedings of Istitute of Mathematics and Mechanics, 23:4 (2017), 32–42 (In Russ.) | MR

[4] Aleev R.Zh., “Units of character fields and central units of integer group rings of finite groups”, Siberian Advances of Mathematics, 11:1 (2001), 1–33 | MR | MR | Zbl | Zbl

[5] R. Ž. Aleev, “Higman's central unit theory, units of integer group rings of finite cyclic groups and Fibonacci numbers”, International Journal of Algebra and Computation, 4:3 (1994), 309–358 | DOI | MR | Zbl

[6] Aleev R.Zh., “Central elements of integer group rings”, Algebra and Logic, 39:5 (2000), 293–300 | DOI | MR | Zbl

[7] Aleev R.Zh., Mukhamadeeva I.R., “Computation of quantum factorials and their inverses”, Chelyabinsk Physical and Mathematical Journal, 1:1 (2016), 6–15 (In Russ.) | MR

[8] Van der Varden B.L., Algebra, 2, Nauka Publ., Moscow, 1979, 624 pp. (In Russ.) | MR

[9] J. J. Liang, “On the integer basis of the maximal real subfield of a cyclotomic field”, Journal für die reine und angewandte Mathematik, Band 286/287 (1976), 223–226 | MR | Zbl

[10] The GAP Group, GAP — Groups, Algorithms, and Programming, Version 4.7.7, 2015 (accessed 15.06.2018) http://www.gap-system.org