Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on arithmetic progressions modular forms.

In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod a
 b
 , i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod a
 b
 through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains.

background

https://www.mdpi.com/2227-7390/5/4/82/pdf

A Course in Arithmetic

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

The generalized subrings of \( \mathbb{F}_q \) are classified, the generalized subrings of ℤ/p2 are investigated, and their complete classification is obtained for p = 2. Examples of \( \mathbb{F}_\infty \)-similar generalized fields, a computation of \( \mathbb{F}_\infty ^{ \otimes n} \), a description of cofinite subrings of ℤp, and examples of subrings of ℤ∞ are given. A conjecture concerning cofinite subrings of ℤ is proposed, and some evidence for it is presented. Bibliography: 4 titles.

Generalized subrings of arithmetic rings

This chapter discusses local fields. Local behavior of a 1-dimensional scheme X near a “nice” point x is described by the local ring, whose completion is a complete discrete valuation ring with residue field k ( x ). The class of complete discrete valuation fields is closely connected with global fields—algebraic number and rational function fields. Local class field theory is one of the highest tops of classical algebraic number theory. It establishes a 1–1 correspondence between abelian extensions of a complete discrete valuation field F whose residue field is finite and subgroups in the multiplicative group F *. In the equal-characteristic cases for an arbitrary field K , there exists a complete discrete valuation field F , whose residue field is isomorphic to K . For the unequal-characteristic case: if K is a field of characteristic p , then there is a complete discrete valuation field F of characteristic 0 with prime element p and residue field K .

Complete Discrete Valuation Fields. Abelian Local Class Field Theories

simplicial complexes with a topology in a natural manner. This homology functor has properties which allow deriving important basic theorems about such spaces. Suppose S is an abstract simplex (i.e., a finite set). Let S be the set of formal linear combinations ∑ v∈S αvv, where S is a simplex, αv ∈ [0, 1], and ∑ v∈S αv = 1. If the vertices of S are instantiated as points in some Euclidean space, then S is the convex hull; however the abstract definition is more general. If K is an abstract simplicial complex then K equals the union of the S over the simplexes S of K. For S with |S| = n, choosing linearly independent points in Rn for the vertices v ∈ S yields an identification of S with a subset of Rn, by which S may be given a topology; indeed it is a metric space with metric √∑ v∈S(αv − βv). The collection {S} satisfies the hypotheses of theorem 17.5, yielding a topology on K coherent with those of the S, which are closed subspaces. We will call a space of the form K, where K is an abstract simplicial complex, a simplicial complex, although usage varies. Many authors require a simplicial complex to be given as a subset of some Euclidean space. Although we will not prove it, a simplicial complex may be given as a subspace of some Euclidean space iff it is countable, the simplexes have bounded dimension, and each vertex belongs to only finitely many simplexes (see [Spanier]). The map K 7→ K is readily seen to be the object map of a functor from ASC to Top. A simplicial map f : K 7→ L induces a continuous function f : K 7→ L by linearity, that is, f(i αvv) = ∑ i αvf(v). This is

Introductory Algebra, Topology, and Category Theory

In this paper, we discuss idempotent elements in the finite ring H/ Zp. The number of idempotents in H/Zp was found recently in [4]. We provide examples and we establish conditions for idempotency in H/Zp. Mathematics Subject Classification: 15A33, 15A30, 20H25, 15A03

A Course in Arithmetic

Citations

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

Generalized subrings of arithmetic rings

Complete Discrete Valuation Fields. Abelian Local Class Field Theories

Introductory Algebra, Topology, and Category Theory

Idempotent Elements in Quaternion Rings over Zp

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