A Concrete Introduction to Higher Algebra, 2nd Edition [Paperback]

Lindsay N. Childs (Author)

Book Description

ISBN-10: 0387989994 | ISBN-13: 978-0387989990 | Publication Date: January 14, 2000 | Edition: 2nd

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, with much emphasis placed on congruence classes leading the way to finite groups and finite fields. New examples and theory are integrated in a well-motivated fashion and made relevant by many applications -- to cryptography, coding, integration, history of mathematics, and especially to elementary and computational number theory. The later chapters include expositions of Rabiin's probabilistic primality test, quadratic reciprocity, and the classification of finite fields. Over 900 exercises, ranging from routine examples to extensions of theory, are scattered throughout the book, with hints and answers for many of them included in an appendix.

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 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 an alternate Paperback edition.

Product Details

• Paperback: 522 pages
• Publisher: Springer; 2nd edition (January 14, 2000)
• Language: English
• ISBN-10: 0387989994
• ISBN-13: 978-0387989990
• Product Dimensions: 9.1 x 6.1 x 1.1 inches
• Shipping Weight: 1.7 pounds (View shipping rates and policies)
• Average Customer Review: 4.4 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #390,069 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.