Arne Ledet

Bio: Arne Ledet is an academic researcher from Texas Tech University. The author has contributed to research in topics: Galois group & Galois extension. The author has an hindex of 7, co-authored 24 publications receiving 365 citations. Previous affiliations of Arne Ledet include Queen's University.

Generic Polynomials: Constructive Aspects of the Inverse Galois Problem

Abstract: 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.

Brauer Type Embedding Problems

Abstract: Galois theory Inverse Galois theory and embedding problems Brauer groups Group cohomology Quadratic forms Decomposing the obstruction Quadratic forms and embedding problems Reducing the embedding problem Pro-finite Galois theory Bibliography Index.

On the Essential Dimension of p -Groups

Abstract: We improve the known bounds on the essential dimension of p-groups over (large) fields of characteristic p.

Basic Algebra = 代数学入門

Abstract: Determine the value of the variable in each equation.

Essential dimension of finite p -groups

Abstract: We prove that the essential dimension and p-dimension of a p-group G over a field F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F.

On the descending central sequence of absolute Galois groups

Abstract: Let $p$ be an odd prime number and $F$ a field containing a primitive $p$th root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group $G_F$ of $F$. Namely, the third subgroup $G_F^{(3)}$ in the descending $p$-central sequence of $G_F$ is the intersection of all open normal subgroups $N$ such that $G_F/N$ is $1$, ${\Bbb Z}/p^2$, or the extra-special group $M_{p^3}$ of order $p^3$ and exponent $p^2$.

Essential dimension: a survey

Abstract: In the paper we survey research on the essential dimension. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups. We present self-contained proofs of these cases and give applications in the theory of simple algebras and quadratic forms.

Commutative Algebra: Constructive Methods : Finite Projective Modules

Abstract: 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.