# R. Zh. Aleev, O.V. Mitina, T.A. Khanenko. Description of the unit group of the integral group ring of a cyclic group of order 16 ... P. 32-42

The paper is devoted to the description of the group of units of the integral group ring of a cyclic group of order $16$. The groups of units of the integral group rings of cyclic groups of orders $2$ and $4$ are trivial, and the group of units of the integral group ring of a cyclic group of order $8$ is well known. Thus, the case of a cyclic group of order $16$ is the first for which the structure of the group of units of the integral group ring of a cyclic $2$-group has not been studied completely. When the groups of units of the integral group rings of cyclic $2$-groups of orders greater than 16 are studied, it is necessary to have information on the structure of the groups of units of the integral group rings of cyclic $2$-groups of lower orders, in particular, of order $16$. Thus, we can say that the case of the group of order $16$ is the basis for further research. We describe the group of units of the integral group ring of a cyclic group of order $16$ in terms of local units defined by the characters of a cyclic group of order $16$ and by the units of the ring of integers of the cyclotomic field $\mathbf{Q}_{16}$ obtained by adjoining a primitive root of unity of degree $16$ to the field of rational numbers. That is why we study in detail the structure of the group of units of the ring of integers of the cyclotomic field $\mathbf{Q}_{16}$. In addition, we derive important relations between the coefficients of an arbitrary unit of the integral group ring of a cyclic group of order $16$. These relations will obviously serve as patterns and examples for obtaining similar relations in studying the units for the cases of $2$-groups of orders greater than $16$. Finally, we note that one of the generators of the group of units of the integral group ring of a cyclic group of order $16$ is a singular unit defined by two units of the ring of integers of the cyclotomic field $\mathbf{Q}_{16}$. This unit is the product of the two local units, each of which is not contained in the integral group ring of a cyclic group of order $16$.

Keywords: cyclic group, group ring, unit of a group ring, cyclotomic field, ring of integers of a field, unit of the ring of integers of a cyclotomic field, integral group ring.

The paper was received by the Editorial Office on October 13, 2017

Rifkhat Zhalyalovich Aleev, Dr. Phys.-Math. Sci., Prof., South Ural State University, Chelyabinsk
State University, Chelyabinsk, 454080 Russia, e-mail: aleevrz@susu.ru, aleev@csu.ru .

Ol’ga Viktorovna Mitina, Сand. Sci. (Phys.-Math.), South Ural State University, Chelyabinsk State
University, Chelyabinsk, 454080 Russia, e-mail: ovm@csu.ru .

Tat’yana Aleksandrovna Khanenko, student Chelyabinsk State University, Chelyabinsk, 454080
Russia, e-mail: tanja_1110_94@mail.ru .

REFERENCES

1.   Aleev R.Zh., Mitina O.V., Khistenko E.A. Congruence modulo 2 of circular units in the fields $Q_{16}$ and $Q_{32}$. Chelyab. Fiz.-Mat. Zh., 2016, vol. 1, no. 4, pp. 8-29 (in Russian).

2.   Aleev R.Zh., Mitina O.V., Khanenko T.A. Finding of units for integer group rings of orders 16 and 32 cyclic groups. Chelyab. Fiz.-Mat. Zh., 2016, vol. 1, no. 4, pp. 30-55 (in Russian).

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5.   Aleev R. Zh. Units of character fields and central units  of integral group rings of finite groups. Siberian Advances of Mathematics, 2001, vol. 11, no. 1,  pp. 1-33.