Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive viewpoint in commutative algebra and studies algorithmic approaches alongside several abstract classical theories. Indeed, it revisits these traditional topics with a new and simplifying manner, making the subject both accessible and innovative. The algorithmic aspects of such naturally abstract topics as Galois theory, Dedekind rings, Prfer rings, finitely generated projective modules, dimension theory of commutative rings, and others in the current treatise, are all analysed in the spirit of the great developers of constructive algebra in the nineteenth century. This updated and revised edition contains over 350 well-arranged exercises, together with their helpful hints for solution. A basic knowledge of linear algebra, group theory, elementary number theory as well as the fundamentals of ring and module theory is required. Commutative Algebra: Constructive Methods will be useful for graduate students, and also researchers, instructors and theoretical computer scientists.

Commutative Algebra: Constructive Methods : Finite Projective Modules

We provide a survey of past research and a list of open problems regarding central simple algebras and the Brauer group over a field, intended both for experts and for beginners.

/pdf/open-problems-on-central-simple-algebras-3hck9i3eox.pdf

Open problems on central simple algebras

Let p be a prime number, let K be a field of characteristic not p, containing the p-th roots of unity, and let r≥1 be an integer. We compute the essential dimension of ℤ/p
 r
 ℤ over K (Theorem 4.1). In particular, i) We have edℚ(ℤ/8ℤ)=4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished). ii) We have edℚ(ℤ/p
 r
 ℤ)≥p
 r-1.

/pdf/on-the-essential-dimension-of-cyclic-p-groups-3uuao2jvgd.pdf

On the essential dimension of cyclic p-groups

https://www.math.ubc.ca/~reichst/can-dim.pdf

On the notion of canonical dimension for algebraic groups

We define and study a new numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in P^{n-1}.

This book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is isomorphic to G. Further, if there is such a Galois extension, find an explicit polynomial over K whose Galois group is the prescribed group G. The main theme of the book is an exposition of a family of �generic� polynomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polynomials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of �generic dimension� to address the problem of the smallest number of parameters required by a generic polynomial.

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Papers

Generic Polynomials: Constructive Aspects of the Inverse Galois Problem

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Papers

Generic Polynomials: Constructive Aspects of the Inverse Galois Problem

典子由井