# A Concrete Introduction to Higher Algebra

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Part of the Undergraduate Texts in Mathematics book series (UTM)

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- Editors
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Part of the Undergraduate Texts in Mathematics book series (UTM)

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

algebra field finite group homomorphism matrices number theory

- DOI https://doi.org/10.1007/978-0-387-74725-5
- Copyright Information Springer New York 2009
- Publisher Name Springer, New York, NY
- eBook Packages Mathematics and Statistics
- Print ISBN 978-0-387-74527-5
- Online ISBN 978-0-387-74725-5
- Series Print ISSN 0172-6056
- Buy this book on publisher's site

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