Groups, Rings and Fields (Springer Undergraduate Mathematics Series) Review

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Groups, Rings and Fields (Springer Undergraduate Mathematics Series) ReviewD. A. R. Wallace's Groups, Rings and Fields is a clearly written, carefully constructed, and well-motivated abstract algebra text that is suitable for a one semester introductory course or self-study. It is accessible to readers who have completed the equivalent of an introductory course given in the first year of a British pure mathematics program such as that offered by Geoff Smith's Introductory Mathematics: Algebra and Analysis (Springer Undergraduate Mathematics Series). Building upon some basic number theory, Wallace demonstrates why mathematicians are interested in the algebraic structures in the title. He carefully develops the properties of these structures, using copious and detailed examples to illustrate the definitions and theorems introduced in the text. Wallace also explains why these ideas are important. The reader can gain a clear understanding of the basic concepts of abstract algebra by reading Wallace's exposition and working the well-chosen exercises, to which he provides solutions or outlines of solutions in an appendix.
The text begins with an introductory chapter that covers sets, mappings, equivalence relations, well-ordering, mathematical induction, and countable sets. This provides the logical foundation for the rest of the text, so it must be understood before proceeding to the subsequent material.
Before introducing the algebraic structures mentioned in the title, Wallace discusses some basic number theory, including divisibility, common divisors, the Division Algorithm, the Euclidean Algorithm, and primes. He then shows that these same concepts apply to polynomials. Wallace explains that when mathematicians recognize the importance of such common properties, they construct an axiomatic system in which the properties are satisfied. In this case, the structure mathematicians constructed is a ring.
Wallace defines a ring and explains the importance of the axioms used in its definition. While Wallace gives some examples of rings and introduces some basic definitions, he defers the discussion of ring theory until after he has discussed groups, which have a simpler, if less familiar, structure.
Wallace introduces semigroups and monoids before defining a group, successively introducing additional axioms until the definition of a group is complete. Since the examples of semigroups and monoids given in the text are not mathematically robust, I assume he did this in order to demonstrate how the properties of groups are derived from the axioms. Once the definition of a group is established and several examples are introduced, Wallace discusses finite and infinite groups, subgroups, conjugacy, normal subgroups, cosets, Lagrange's Theorem, factor-groups (which are called quotient groups in some texts), group homomorphisms, and the isomorphism theorems for groups.
Wallace begins his discussion of ring theory with modular arithmetic, using the ring of integers together with the ring of integers modulo n to distinguish between a ring, an integral domain, and a field. After illustrating these concepts with numerous examples, Wallace introduces Euclidean domains, ideals, ring homomorphisms, the isomorphism theorems for rings, principal ideal domains, and unique factorization domains, while explaining the relationships between them. He also briefly discusses factoring polynomials with rational coefficients.
Wallace concludes the text with some additional topics in group theory, including permutation groups, generators and relations, direct products and sums, Abelian groups, and the Sylow theorems. The latter part of the chapter is noticeably harder to read than the rest of the book because the material is more abstract and the proofs are more complex.
If you plan to do graduate work in mathematics, you will need a more comprehensive text. Wallace recommends John B. Fraleigh's A First Course in Abstract Algebra, 7th Edition and Joseph Gallian's Contemporary Abstract Algebra for further study. A more ambitious student could work through Michael Artin's Algebra, David Dummitt and Richard Foote's Abstract Algebra, or I. N. Herstein's Topics in Algebra. Another option would be to use this text as preparation for Geoff Smith and Olga Tabachnikova's Topics in Group Theory (Springer Undergraduate Mathematics Series).
A caveat is that the exposition is marred by numerous misprints, a few of which are in the answer key. For instance, the exponents in the explicit formula given for the Fibonacci sequence in part (ii) of exercise 11 in section 1.5 are incorrect. They should be n's rather than n + 1's. The other errors are easier to detect.
Finally, since this is a British text, American readers will have to adjust to differences in terminology (prime-pair instead of twin prime) and notation (what looks like a decimal point to American readers actually means multiplication).Groups, Rings and Fields (Springer Undergraduate Mathematics Series) Overview

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