卷 234, 编号 2 (2018)

The reduced group of principal units of cyclic extensions of local fields is considered as a Galois module. This module is studied by examining the Jordan canonical form of the generating automorphism of the Galois group.

Let \( \mathcal{C} \) be an additive category in which each morphism has a kernel. It is proved that the homotopy category of the category of complexes over \( \mathcal{C} \) which are concentrated in degrees 2, 1, 0 and are exact in degrees 2 and 1 is Abelian. Under assumption that a category \( \mathcal{C} \) is Abelian, this result was obtained earlier by considering the heart of a suitable t-structure on the homotopy category of \( \mathcal{C} \).

The aim of the paper is to construct an embedding of a given cyclic extension of degree p of a complete discrete valuation field of characteristic 0 with an arbitrary residue field of characteristic p > 0 into a cyclic extension of degree p². The result extends the construction obtained by S. V. Vostokov and I. B. Zhukov in terms of Witt vectors, to a wider interval of values for the ramification jump of the original field extension.

An elementary necessary condition is established for a prime integer of the form 8k + 1 to be congruent.

An association scheme is said to be T-amorphous if it is antisymmetric and any tournament obtained by an appropriate merging of its classes is doubly regular. The goal of this paper is to study basic properties of T-amorphous schemes.

The restrictions of irreducible representations of the special linear group over an algebraically closed field of positive characteristic p to subsystem subgroups of type A₂ are studied. The composition factors for such restrictions are described in the case where the highest weights of the representations are locally small.

Generalized subrings of the ring of integers that give birational models for the field of rationals are studied. A homogeneous strengthening of Evdokimov’s theorem is proved. An approach to calculation of homotopy groups with the help of generalized rings is proposed.

Let R = \( \prod \limits_{i\in I}{F}_i \) be the direct product of fields, and let\( S=R\left[\sqrt{d}\right]=\prod \limits_{i\in I}{F}_i\left[\sqrt{d_i}\right] \) be a rank 2 extension of R. The subgroups of the general linear group GL(2n,R), n ≥ 3, that contain the elementary group E (n, S) are described. It is shown that for every such a subgroup H there exists a unique ideal A ⊴ R such that E (n, S)E(2n,R,A) ≤ H ≤ N_GL(2n,R) (E (n, S)E(2n,R,A)).