The CRC Handbook of Finite Fields (hereafter referred to as the Handbook) is a reference
book for the theory and applications of finite fields. It is not intended to be an introductory …

These lecture notes were prepared by the authors for use in graduate courses and
seminars, based on the work of many earlier mathematicians. In addition to very elementary …

The main classes of finite rings (fr) and modules interesting for applications are considered:
Wedderburn rings, local, chain and Galois rings,(quasi)-Frobenius bimodules. Polynomials …

Starting from the classical finite-dimensional Galois theory of fields, this book develops
Galois theory in a much more general context, presenting work by Grothendieck in terms of …

Contents. 1. Introduction and Notation; 2. A Jacobian Criterion for Separability; 3. A Jacobian
Criterion for Extension of Derivations; 4. Explicit Formulas for the Separability Idempotent; …

This book presents a comprehensive introduction to the theory of separable algebras over
commutative rings. After a thorough introduction to the general theory, the fundamental roles …

Rings of Order p^{5} Part I. Nonlocal Rings Page 1 Ž . Journal of Algebra 231, 677690 2000
doi:10.1006rjabr.2000.8349, available online at http:rrwww.idealibrary.com on Rings of Order …

A chain ring is an associative, commutative ring with an identity whose ideals form a chain.
We associate with each finite chain ring five invariants (integers) and determine (in certain …

In this paper we examine finite local commutative rings which are the building blocks of finite
commutative rings. For a finite local ring R our attention centers on the basic structural …

We prove that every finite p-ring R contains a unique (up to isomorphism) subring S such
that $ S/pS\cong R/{\operatorname {rad}}\; R $. S is shown to be a direct sum of full matrix …