Class Groups Of Cyclotomic Fields Research Articles

On minus quotients of ideal class groups of cyclotomic fields

Let [Formula: see text] be the minus quotient of the ideal class group of the [Formula: see text]th cyclotomic field. In this paper, first, we show that each finite abelian group appears as a subgroup of [Formula: see text] for some [Formula: see text]. Second, we show that, for all pairs of integers [Formula: see text] and [Formula: see text] with [Formula: see text], the kernel of the lifting map [Formula: see text] is contained in the [Formula: see text]-torsion [Formula: see text] of [Formula: see text]. Such an evaluation of the exponent is an individuality of cyclotomic fields.

On some exact sequences in the theory of cyclotomic fields

The purpose of this article is to show the existence of some exact sequences which relate the ideal class groups of cyclotomic fields to Gauss sums. These exact sequences imply the results of Hachimori, Ichimura and Beliaeva. Furthermore, we study the relations between the étale cohomology groups and some conjectures on cyclotomic fields.

Cubic structures and ideal class groups

We establish a generalization of Breen's theory of cubic structures on line bundles over group schemes. We study such “ n-cubic structures” inductively using multiextensions. As a result we obtain information on the set of isomorphism classes of line bundles with n-cubic structures over finite multiplicative group schemes over Spec ( Z ) by relating this set to certain corresponding eigenspaces of ideal class groups of cyclotomic fields.

Cyclotomic Integers of Prescribed Absolute Value and the Class Group

We obtain a new method for the study of class groups of cyclotomic fields by investigating cyclotomic integers of prescribed absolute value. Explicit subgroups of the classgroupCmodulo the class groupC+of the maximal real subfield are exhibited and lower bounds on their orders are derived. For themth cyclotomic fieldKm, wherem=pam′, (p, m′)=1, andpis a prime, we determine the structure ofC+CP/C+CQup to a binary parameter; hereCP,CQare the subgroups ofCgenerated by the classes [Pi] respectively [Qi], wherepfactors inKmas ∏Qi,Qi=Pϕ(pa)i, and thePiare prime ideals.

An extension theorem for integral representations

AbstractBy a fundamental theorem of Brauer every irreducible character of a finite groupGcan be written in the field Q(εm) ofmth roots of unity wheremis the exponent ofG. Is it always possible to find a matrix representation over its ring Z[εm] of integers? In the present paper it is shown that this holds true provided it is valid for the quasisimple groups. The reduction to such groups relies on a useful extension theorem for integral representations. Iwasawa theory on class groups of cyclotomic fields gives evidence that the answer is at least affirmative when the exponent is replaced by the order, and provides for a general qualitative result.

On the Exponents of Ideal Class Groups of Cyclotomic Fields

On the Exponents of Ideal Class Groups of Cyclotomic Fields

Ideal Class Groups of Cyclotomic Fields and Modular Forms of Level 1

The relations between the conjecture about the minus part of the ideal class group of the cyclotomic field Q(μ p ) (which is a consequence of the Vandiver conjecture) and the structures of the Hecke algebra and of the Eisenstein ideal of level 1 modular forms are studied.

On the exponents of ideal class groups of cyclotomic fields

It will be proved that the ideal class group of the cyclotomic field of 65th roots of unity is of type ( 4 , 4 , 2 , 2 ) (4,4,2,2) , and remarks on the exponents of ideal class groups of cyclotomic fields will be made.

Cohomology of class groups of cyclotomic fields; an application to Morse-Smale diffeomorphisms

The obstruction group SSF came up in the topological investigations of Shub and Franks. It was studied by Bass and Lenstra; the latter showed that SSF ≅ ⊕ n⩾1 Pic Z ζ n, 1 n . In this paper the structure of SSF as an abelian group is determined. It is proved that SSF ≅ ⊕ n⩾1 Z⧸/n Z confirming a conjecture of Lenstra.

Class Groups of Cyclotomic Fields and Group Rings

Class Groups of Cyclotomic Fields and Group Rings