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A Guide to Groups, Rings, and Fields

A Guide to Groups, Rings, and Fields

Fernando Q. Gouvêa
Volume: 48
Copyright Date: 2012
Edition: 1
Pages: 328
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  • Book Info
    A Guide to Groups, Rings, and Fields
    Book Description:

    This Guide offers a concise overview of the theory of groups, rings, and fields at the graduate level, emphasizing those aspects that are useful in other parts of mathematics. It focuses on the main ideas and how they hang together. It will be useful to both students and professionals. In addition to the standard material on groups, rings, modules, fields, and Galois theory, the book includes discussions of other important topics that are often omitted in the standard graduate course, including linear groups, group representations, the structure of Artinian rings, projective, injective and flat modules, Dedekind domains, and central simple algebras. All of the important theorems are discussed, without proofs but often with a discussion of the intuitive ideas behind those proofs. Those looking for a way to review and refresh their basic algebra will benefit from reading this Guide, and it will also serve as a ready reference for mathematicians who make use of algebra in their work.

    eISBN: 978-1-61444-211-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. I-VI)
  2. Table of Contents
    (pp. VII-XIV)
  3. Preface
    (pp. XV-XVIII)
    Fernando Q. Gouvêa
  4. A Guide to this Guide
    (pp. 1-2)
  5. CHAPTER 1 Algebra: Classical, Modern, and Ultramodern
    (pp. 3-8)

    The word “algebra” is derived from the title of a famous book by Baghdadi mathematician Muḥammad Ibn Mūsa Al-Khwārizmī, who flourished in the 9th century. His name indicates that he (or his family) was from Khwārizm, a town south of the Aral Sea in what is now Uzbekistan. (It is currently called Khiva.) Among Al-Khwārizmī’s influential books was one on “aljabr w’al-muqābala,” which means something like “restoration and compensation.”

    The book starts off with quadratic equations, then goes on to practical geometry, simple linear equations, and a long discussion of how to apply mathematics to solve inheritance problems. The portion...

  6. CHAPTER 2 Categories
    (pp. 9-16)

    From the standpoint of category theory, all of mathematics is about objects and arrows: groups and homomorphisms, topological spaces and continuous functions, differentiable manifolds and smooth maps, etc. This gives a useful way of thinking about various mathematical theories, but more importantly it highlights connections between different theories, such as going from a topological space to its first homology group. Since categories are about objects and arrows, one expects functors to map objects to objects andarrows to arrows. It is the latter which turns out to be the fundamental insight: “functorial” constructions are important.

    For our purposes, category theory...

  7. CHAPTER 3 Algebraic Structures
    (pp. 17-28)

    Anoperation(more precisely, abinary operation) on a setSis a function fromS × StoS. Standard examples are addition, multiplication, and composition of functions.

    Elementary texts often emphasize the “closure” property of an operation (or, sometimes, of an algebraic structure): the product of two elements inSmust be an element ofS. We have, instead, built this into the definition.

    Analgebraic structure(Bourbaki says amagma) is a set equipped with one or more operations. Such structures sometimes come with distinguished elements (such as identity elements) or functions associated with the operation (such...

  8. CHAPTER 4 Groups and their Representations
    (pp. 29-106)

    At first, groups were groups of transformations. In the theory of equations, they appeared permuting the roots of polynomials and permuting the variables in rational functions. A few decades after, groups of geometric transformations were discovered and studied. It was only much later that the abstract notion of a group was introduced.

    A modern approach must start from the abstract notion. But history reminds us to introduce group actions early in the game and to study groups via their actions. This is how we present the theory here.

    The crucial definitions specify the objects, the acceptable functions, and what an...

  9. CHAPTER 5 Rings and Modules
    (pp. 107-220)

    Rings may well be the most familiar algebraic structure. We all grew up with integers, polynomials, rational and real numbers. These familiar rings do not, however, prepare us for the huge variety of rings and the complexity of ring theory. Rings and their modules should be studied together, and that is what we do in this chapter.

    We start fromthe definitions of the objects, the appropriate homomorphisms, and the relevant sub-objects. Since both rings and modules will be in play, we need to do this for both structures.

    Definition 5.1.1A ring is a set R together with two operations...

  10. CHAPTER 6 Fields and Skew Fields
    (pp. 221-276)

    This chapter studies the theory of division rings, i.e., fields and skew fields. To do that, we deploy both group theory and ring theory as developed in the previous chapters.

    We begin by repeating the definitions and setting up some standard notations.

    Definition 6.1.1 A division ringis a ring in which0 ≠ 1and every nonzero element has a multiplicative inverse. A noncommutative division ring is called askew field. Acommutative division ring is called afield.

    The choice of the word “field” seems to be peculiar to English; in other European languages, the word chosen is...

  11. Bibliography
    (pp. 277-282)
  12. Index of Notations
    (pp. 283-294)
  13. Index
    (pp. 295-308)
  14. About the Author
    (pp. 309-309)