Описание Группы Единиц Целочисленного Группового Кольца Циклической Группы Порядка 16

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

doi:10.21538/0134-4889-2017-23-4-32-42

Cited by 1 publication

(1 citation statement)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Информацию о нём можно найти либо в диссертации [3], либо в [14]. Случай 2 4 = 16 полностью изучен в работах [5] и [6].…”

Section: аннотацияunclassified

Units of integral group rings of cyclic $2$-groups

Aleeev,

Mitina,

Godova

2021

Preprint

View full text Add to dashboard Cite

This paper is devoted to the units of integral group rings of cyclic 2groups of small orders, namely, the orders of 2 n for n 7. Immediately we should note the issues our consideration describe in the introduction in more detail.Here we will indicate the main directions of our research. Previously, we proved that the normalized group of units of an integral group ring of a cyclic 2-group of order 2 n contains a subgroup of finite index, which is the direct product of the subgroup of units defined by the character with the largest character field and the subgroup of units that is isomorphic to the subgroup of units of the integer group ring of the cyclic 2-group of order 2 n−1 . Because of this, it is very important to study the structure of the subgroup of units defined by the character with the largest field of characters, which is the cyclotomic field Q 2 n obtained by adjoining a primitive 2 n th root of unity to Q, the field of rational number. That subgroup of units of an integral group ring of a cyclic 2-group is isomorphic to the subgroup of the group of units of the integer ring of the specified cyclotomic field. Therefore, the research of units of an integer group ring of a cyclic 2-group is reduced to study the properties of the group of units of the integer ring of the cyclotomic field Q 2 n . In general, the group of units of the ring of integers of the circular field Q 2 n is not known completely. However, the fundamental paper of Sinnott allows us to always find a subgroup of circular units of a finite index in this group of units, and this index is equal to the class number of the maximum real subfield of the field Q 2 n . The classical Weber class number problem assumes the class number is equal to 1, and this would give a coincidence of the subgroup of circular units and the group of all units of the integer ring of the cyclotomic field Q 2 n . As far as we know, the Weber problem is solved positively for all 2 n 256, and under the generalized Riemann hypothesis for 2 n 512.Thus, we will study of groups of circular units of integer rings of cyclotomic fields Q 2 n in large part. First of all, we note one significant

show abstract

Section: аннотацияunclassified