A Concrete Introduction to Higher Algebra (Undergraduate Texts in Mathematics) [Hardcover]

Lindsay N. Childs (Author)

Book Description

ISBN-10: 0387944842 | ISBN-13: 978-0387944845 | Publication Date: October 19, 1995 | Edition: 2nd

This book is written as an introduction to higher algebra for students with a background of a year of calculus. The objective of the book is to give students enough experience in the algebraic theory of the integers and polynomials to appreciate the basic concepts of abstract algebra. The main theoretical thread is to develop algebraic properties of the ring of integers: unique factorization into primes, congruences and congruence classes, Fermat's theorem, the Chinese remainder theorem, and then again for the ring of polynomials.
Concurrently with the theoretical development, the book presents a broad variety of applications, to cryptography, error-correcting codes, Latin squares, tournaments, techniques of integration and especially to elementary and computational number theory. Many of the recent advances in computational number theory are built on the mathematics which is presented in this book. Thus the book may be used as a first course in higher algebra, as originally intended, but may also serve as an introduction to modern computational number theory, or to applied algebra.

Editorial Reviews

Review

From the reviews: "The user-friendly exposition is appropriate for the intended audience. Exercises often appear in the text at the point they are relevant, as well as at the end of the section or chapter. Hints for selected exercises are given at the end of the book. There is sufficient material for a two-semester course and various suggestions for one-semester courses are provided. Although the overall organization remains the same in the second edition¿Changes include the following: greater emphasis on finite groups, more explicit use of homomorphisms, increased use of the Chinese remainder theorem, coverage of cubic and quartic polynomial equations, and applications which use the discrete Fourier transform." MATHEMATICAL REVIEWS From the reviews of the third edition: "This book can serve as both an introduction to number theory and abstract algebra, sacrifices have to be made with respect to its algebraic content. … the book has been written with a high degree of rigor and accuracy and I definitely recommend it for consideration as the basis of an alternative route into abstract algebra and its applications." (The Mathematical Association of America, April, 2009) "The target audience remains students requiring a substantial introduction to the elements of university-level algebra. … the text proceeds throughout on a foundation built from the students’ familiarity with integers and polynomials over fields. Great care is taken to proceed to abstract concepts by way of familiar examples, and a great many exercises are provided throughout the text. … A noteworthy feature of the book is the inclusion of extensive material on applications, to such topics as cryptography and factoring polynomials." (Kenneth A. Brown, Mathematical Reviews, Issue 2009 i) --This text refers to the Paperback edition.

From the Back Cover

This book is an informal and readable introduction to higher algebra at the post-calculus level. The concepts of ring and field are introduced through study of the familiar examples of the integers and polynomials. A strong emphasis on congruence classes leads in a natural way to finite groups and finite fields. The new examples and theory are built in a well-motivated fashion and made relevant by many applications - to cryptography, error correction, integration, and especially to elementary and computational number theory. The later chapters include expositions of Rabin's probabilistic primality test, quadratic reciprocity, the classification of finite fields, and factoring polynomials over the integers. Over 1000 exercises, ranging from routine examples to extensions of theory, are found throughout the book; hints and answers for many of them are included in an appendix. The new edition includes topics such as Luhn's formula, Karatsuba multiplication, quotient groups and homomorphisms, Blum-Blum-Shub pseudorandom numbers, root bounds for polynomials, Montgomery multiplication, and more. "At every stage, a wide variety of applications is presented...The user-friendly exposition is appropriate for the intended audience" - T.W. Hungerford, Mathematical Reviews "The style is leisurely and informal, a guided tour through the foothills, the guide unable to resist numerous side paths and return visits to favorite spots..." - Michael Rosen, American Mathematical Monthly --This text refers to the Paperback edition.

Product Details

• Hardcover: 522 pages
• Publisher: Springer; 2nd edition (October 19, 1995)
• Language: English
• ISBN-10: 0387944842
• ISBN-13: 978-0387944845
• Product Dimensions: 9 x 6.6 x 0.8 inches
• Shipping Weight: 1.4 pounds
• Average Customer Review: 4.4 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #3,093,633 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

11 of 11 people found the following review helpful:

4.0 out of 5 stars A pleasure to read, June 3, 2003

By 

Hasnor Lot (Minneapolis) - See all my reviews

This review is from: A Concrete Introduction to Higher Algebra (Undergraduate Texts in Mathematics) (Hardcover)

Let it be proclaimed as the first axiom of mathematics writing: Difficulty should never be multiplied without necessity. The informal prose is the one feature that stands out prominently throughout this book.

The misty road towards a true understanding of the abstract and sometimes difficult concepts in modern algebra (group, ring, field, homomorphism etc) is paved with concrete examples and applications from number theory. This approach not only parallels the very historical development of the subject, but also has the pedagogical advantage that it does not divorce the theory from the practice. Laid to rest is the mythical belief that the whole edifice of modern algebra rests precariously on a theoretical foundation of purely axiomatic constructs separate from everyday reality: any student who can tell time on an analog clock or values the security of her credit card will appreciate the practical contribution of the field.

My only complaint is that the book is rife with errors, of which some can be checked against the errata available on the author's website. Other than that, I lavish only praise upon it.

10 of 10 people found the following review helpful:

5.0 out of 5 stars Introductory but very clear algebra book, April 30, 2008

By 

A reader (Geneva, Switzerland) - See all my reviews

This review is from: A Concrete Introduction to Higher Algebra, 2nd Edition (Paperback)

This is an introductory level textbook in number theory and higher algebra. I particularly liked the extremely clear language and style of the author. He explains most of the passages first in words and then in formulas, making all steps much less abstract than other algebra books tend to do.

I would recommend this especially for self-study, as the book reads exactly as a good teacher talks to a class.

Concerning the errata, this does not bother me. Go to the web site of the author (http://math.albany.edu/~lc802/), download the errata and in ten minutes you will have penciled in the corrections.

9 of 9 people found the following review helpful:

4.0 out of 5 stars Good text and good for the library, June 9, 2000

By 

James M. Cargal (Montgomery, AL USA) - See all my reviews
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This review is from: A Concrete Introduction to Higher Algebra (Undergraduate Texts in Mathematics) (Hardcover)

This book is an introduction to abstract algebra. Essentially Professor Childs gives a minicourse on number theory as a precursor to abstract algebra. He then developes ring theory before group theory and takes a particularly historical approach to groups. There is a lot of serious attention to applications such as coding and cryptography (although both concern "codes" they are different areas). There is attention to finite fields and the foundations of Galois theory are given if not the full treatment. This book is a good text if not a great one, it has enough material to give the instructor flexibility in the course. It is a good book for the serious student to have in his/her library.