Description of the unit group of the integral group ring of a cyclic group of order $16$

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 $16$.