Scratchpad's view of algebra I: Basic commutative algebra

2005

Dear Reader, why did we begin the foreword of this second volume with the same quote as the first? There we wrote that it took three years of intense work just to fill three centimeters of your bookshelf. The completion of this volume took four years and it is about four centimeters thick. Thus we have a confirmed invariant which governs our writing: our velocity is one centimeter per year, after all e↵ects due to Hofstadter's Law have been taken into account. When we started this project in the last millennium, we planned a book for learning, teaching, reading and, most of all, enjoying the topic at hand. Surely there is no law which says that a mathematical book has to be dull, boring, dry, or tedious. But how do you make it enjoyable?

1991

This paper explains how Scratchpad solves the problem of presenting a categorical view of factorization in unique factorization domains, i.e. a view which can be propagated by functors such as SparseUnivari-atePolynomial or Fraction. This is not easy, as the constructive version of the classical concept of Unique-FactorizationDomain cannot be so propagated. The solution adopted is based largely on Seidenberg's conditions (F) and (P), but there are several additional points that have to be borne in mind to produce reasonably efficient algorithms in the required generality. The consequence of the algorithms and interfaces presented in this paper is that Scratchpad can factorize in any extension of the integers or finite fields by any combination of polynomial, fraction and algebraic extensions: a capability far more general than any other computer algebra system possesses. The solution is not perfect: for example we cannot use these general constructions to factorize polynomials in Z[ √ −5][x] since the domain Z[ √ −5] is not a unique factorization domain, even though Z[ √ −5] is, since it is a field. Of course, we can factor polynomials in Z[ √ −5][x].

1980

We describe the basic recursion theory of functions computable by different kinds of register machines designed to operate within an arbitrarily chosen algebraic system. Especial emphasis is placed upon computation in natural algebraic systems such as groups, rings and fields. A useful theorem about the topological structure of computable subsets of a Hausdorff topological algebra is proved. KEY WORDS & PHRASES: algebraic register machines, finite algorithmic procedures, algorithmically decidable and undecidable problems in Algebra, computable functions on algebraic data types *) This report will be published elsewhere and is not for review.

1981

This paper reports ongoing research at th e IBM Research Center on the development of a languag e with extensible parameterized types and generic operator s for computational algebra. The language provides a n abstract data type mechanism for defining algorithm s which work in as general a setting as possible. The language is based on the notions of domains and categories. Domains represent algebraic structures. Categorie s designate collections of domains having common operations with stated mathematical properties. Domain s and categories are computed objects which may b e dynamically assigned to variables, passed as arguments , and returned by functions. Although the language ha s been carefully tailored for the application of algebrai c computation, it actually provides a very general abstrac t data type mechanism. Our notion of a category to group domains with common properties appears novel amon g programming languages (cf. image functor of RUSSELL) and leads to a very powerful notion of abstract algorithm s missing from other work on data types known to th e authors. 1. Introduction. This paper describes a language wit h parameterized types and generic operators particularl y suited to computational algebra. A flexible framewor k is given for building algebraic structures and definin g algorithms which work in as general a setting a s possible. This section will be an overview of our mai n concepts: domains " and " categories In section 2, w e give more precise definitions and some examples. Section 3 contains our conclusions .

Month, 2005

Non-commutative polynomial algebras appear in a wide range of applications, from quantum groups and theoretical physics to linear differential and difference equations.

2014

This is the first edition of a special session at ACA conference devoted to providing a forum for exchange of ideas and research results related to Computer Algebra Aspects of Finite Rings in a broad sense and also to their applications. Session topics will include (but are not limited to) the following: 1) Computer Algebra and Finite Rings Computer representation and computation over finite rings and polynomial rings over finite rings. C.A. software and development of packages devoted to finite rings and related structures ...

Foundations of Computational Mathematics, 2016

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel-Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.

Recursion Theory, its Generalisations and Applications

Notice the instruction can be ]J specified by its characteristic numerical parameters <ao-1 ,]J,cr,>.. 1 , ... ,Ak> where we let ao-1 stand for a number which refers to the kind of the instruction and a stand for that 1 such that a is cr., a point taken up later on in this section when we l consider coding.

2001

Abstract. The use and development of computer technology by algebraists over the last forty years has revolutionised the way in which algebraists think about algebra, and the way they teach it and conduct their research. This paper is a personal reflection on these changes by a somewhat unwilling computer user.