Algebraic Extensions of Fields

2014-01-07
 |  Mathematics
Algebraic Extensions of Fields

Author: Paul J. McCarthy

Publisher: Courier Corporation

ISBN: 9780486781471

Category: Mathematics

Page: 192

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Graduate-level coverage of Galois theory, especially development of infinite Galois theory; theory of valuations, prolongation of rank-one valuations, more. Over 200 exercises. Bibliography. "...clear, unsophisticated and direct..." — Math.

Algebraic Extensions of Fields

1966
 |  Algebraic fields
Algebraic Extensions of Fields

Author: Paul Joseph McCarthy

Publisher:

ISBN: UOM:39015042062219

Category: Algebraic fields

Page: 184

View: 570

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Graduate-level coverage of Galois theory, especially development of infinite Galois theory; theory of valuations, prolongation of rank-one valuations, more. Over 200 exercises. Bibliography. ..."clear, unsophisticated and direct..." -- "Math."

Infinite Algebraic Extensions of Finite Fields

1989
 |  Mathematics
Infinite Algebraic Extensions of Finite Fields

Author: Joel V. Brawley

Publisher: American Mathematical Soc.

ISBN: 9780821851012

Category: Mathematics

Page: 104

View: 620

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Over the last several decades there has been a renewed interest in finite field theory, partly as a result of important applications in a number of diverse areas such as electronic communications, coding theory, combinatorics, designs, finite geometries, cryptography, and other portions of discrete mathematics. In addition, a number of recent books have been devoted to the subject. Despite the resurgence in interest, it is not widely known that many results concerning finite fields have natural generalizations to abritrary algebraic extensions of finite fields. The purpose of this book is to describe these generalizations. After an introductory chapter surveying pertinent results about finite fields, the book describes the lattice structure of fields between the finite field $GF(q)$ and its algebraic closure $\Gamma (q)$. The authors introduce a notion, due to Steinitz, of an extended positive integer $N$ which includes each ordinary positive integer $n$ as a special case. With the aid of these Steinitz numbers, the algebraic extensions of $GF(q)$ are represented by symbols of the form $GF(q^N)$. When $N$ is an ordinary integer $n$, this notation agrees with the usual notation $GF(q^n)$ for a dimension $n$ extension of $GF(q)$. The authors then show that many of the finite field results concerning $GF(q^n)$ are also true for $GF(q^N)$. One chapter is devoted to giving explicit algorithms for computing in several of the infinite fields $GF(q^N)$ using the notion of an explicit basis for $GF(q^N)$ over $GF(q)$. Another chapter considers polynomials and polynomial-like functions on $GF(q^N)$ and contains a description of several classes of permutation polynomials, including the $q$-polynomials and the Dickson polynomials. Also included is a brief chapter describing two of many potential applications. Aimed at the level of a beginning graduate student or advanced undergraduate, this book could serve well as a supplementary text for a course in finite field theory.

Field Extensions and Galois Theory

1984-12-28
 |  Mathematics
Field Extensions and Galois Theory

Author: Julio R. Bastida

Publisher: Cambridge University Press

ISBN: 0521302420

Category: Mathematics

Page: 352

View: 690

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This 1984 book aims to make the general theory of field extensions accessible to any reader with a modest background in groups, rings and vector spaces. Galois theory is regarded amongst the central and most beautiful parts of algebra and its creation marked the culmination of generations of investigation.

A Field Guide to Algebra

2007-12-21
 |  Mathematics
A Field Guide to Algebra

Author: Antoine Chambert-Loir

Publisher: Springer Science & Business Media

ISBN: 9780387269559

Category: Mathematics

Page: 198

View: 336

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This book has a nonstandard choice of topics, including material on differential galois groups and proofs of the transcendence of e and pi. The author uses a conversational tone and has included a selection of stamps to accompany the text.

Foundations of Galois Theory

2014-07-10
 |  Mathematics
Foundations of Galois Theory

Author: M.M. Postnikov

Publisher: Elsevier

ISBN: 9781483156477

Category: Mathematics

Page: 122

View: 625

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Foundations of Galois Theory is an introduction to group theory, field theory, and the basic concepts of abstract algebra. The text is divided into two parts. Part I presents the elements of Galois Theory, in which chapters are devoted to the presentation of the elements of field theory, facts from the theory of groups, and the applications of Galois Theory. Part II focuses on the development of general Galois Theory and its use in the solution of equations by radicals. Equations that are solvable by radicals; the construction of equations solvable by radicals; and the unsolvability by radicals of the general equation of degree n ? 5 are discussed as well. Mathematicians, physicists, researchers, and students of mathematics will find this book highly useful.

Abstract Algebra with Applications

2018-05-04
 |  Mathematics
Abstract Algebra with Applications

Author: Karlheinz Spindler

Publisher: Routledge

ISBN: 9781351469241

Category: Mathematics

Page: 326

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A comprehensive presentation of abstract algebra and an in-depth treatment of the applications of algebraic techniques and the relationship of algebra to other disciplines, such as number theory, combinatorics, geometry, topology, differential equations, and Markov chains.

Handbook of Mathematics

2016-12-07
 |  Mathematics
Handbook of Mathematics

Author: Thierry Vialar

Publisher: BoD - Books on Demand

ISBN: 9782955199008

Category: Mathematics

Page: 1134

View: 810

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The book consists of XI Parts and 28 Chapters covering all areas of mathematics. It is a tool for students, scientists, engineers, students of many disciplines, teachers, professionals, writers and also for a general reader with an interest in mathematics and in science. It provides a wide range of mathematical concepts, definitions, propositions, theorems, proofs, examples, and numerous illustrations. The difficulty level can vary depending on chapters, and sustained attention will be required for some. The structure and list of Parts are quite classical: I. Foundations of Mathematics, II. Algebra, III. Number Theory, IV. Geometry, V. Analytic Geometry, VI. Topology, VII .Algebraic Topology, VIII. Analysis, IX. Category Theory, X. Probability and Statistics, XI. Applied Mathematics. Appendices provide useful lists of symbols and tables for ready reference. The publisher’s hope is that this book, slightly revised and in a convenient format, will serve the needs of readers, be it for study, teaching, exploration, work, or research.

Abstract Algebra

2007-07-21
 |  Mathematics
Abstract Algebra

Author: Pierre Antoine Grillet

Publisher: Springer Science & Business Media

ISBN: 9780387715681

Category: Mathematics

Page: 674

View: 976

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A completely reworked new edition of this superb textbook. This key work is geared to the needs of the graduate student. It covers, with proofs, the usual major branches of groups, rings, fields, and modules. Its inclusive approach means that all of the necessary areas are explored, while the level of detail is ideal for the intended readership. The text tries to promote the conceptual understanding of algebra as a whole, doing so with a masterful grasp of methodology. Despite the abstract subject matter, the author includes a careful selection of important examples, together with a detailed elaboration of the more sophisticated, abstract theories.

Bilinear Algebra

2017-11-22
 |  Mathematics
Bilinear Algebra

Author: Kazimierz Szymiczek

Publisher: Routledge

ISBN: 9781351464208

Category: Mathematics

Page: 413

View: 511

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Giving an easily accessible elementary introduction to the algebraic theory of quadratic forms, this book covers both Witt's theory and Pfister's theory of quadratic forms. Leading topics include the geometry of bilinear spaces, classification of bilinear spaces up to isometry depending on the ground field, formally real fields, Pfister forms, the Witt ring of an arbitrary field (characteristic two included), prime ideals of the Witt ring, Brauer group of a field, Hasse and Witt invariants of quadratic forms, and equivalence of fields with respect to quadratic forms. Problem sections are included at the end of each chapter. There are two appendices: the first gives a treatment of Hasse and Witt invariants in the language of Steinberg symbols, and the second contains some more advanced problems in 10 groups, including the u-invariant, reduced and stable Witt rings, and Witt equivalence of fields.