|
This article is cited in 1 scientific paper (total in 1 paper)
Mathematics
Local units of integer group ring of cyclic group of order 64 for character with character field ${\mathbb Q}_{64}$
R. Zh. Aleevab, O. V. Mitinaba, T. A. Khanenkob a South Ural State University (National Research University), Chelyabinsk, Russia
b Chelyabinsk State University, Chelyabinsk, Russia
Abstract:
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 $64$.
Keywords:
group ring, unit of group ring, cyclic group, cyclotomic field, integer group ring.
Received: 02.06.2018 Revised: 03.08.2018
Citation:
R. Zh. Aleev, O. V. Mitina, T. A. Khanenko, “Local units of integer group ring of cyclic group of order 64 for character with character field ${\mathbb Q}_{64}$”, Chelyab. Fiz.-Mat. Zh., 3:3 (2018), 253–275
Linking options:
https://www.mathnet.ru/eng/chfmj104 https://www.mathnet.ru/eng/chfmj/v3/i3/p253
|
Statistics & downloads: |
Abstract page: | 147 | Full-text PDF : | 104 | References: | 13 |
|