Advanced Number Theory by Harvey Cohn

Product Description

Eminent mathematician/teacher approaches algebraic number theory from historical standpoint. Demonstrates how concepts, definitions, theories have evolved during last 2 centuries. Abounds with numerical examples, over 200 problems, many concrete, specific theorems. Includes numerous graphs and tables.

Product Details

* Amazon Sales Rank: #700713 in Books
* Published on: 1980-08-01
* Number of items: 1
* Binding: Paperback
* 288 pages

Customer Reviews

All about quadratric number fields, but little else4
From 1962, this is a detailed account of quadratic number fields, and makes a fair introduction to the theory of number fields of any degree. Ideal theory (restricted to the quadratic case) is well covered in plenty of detail. Gauss's classic theory of binary quadratic forms is also included.

Cohn is clearly quite keen on the subject, and is not just writing a textbook on some arbitrary topic for which he thinks there might be a market. And he has no fear of including pedagogical remarks in a textbook. The English is a bit awkward in places, but that is a minor thing.

The basics about characters and Dirichlet L-series are developed, but only to the extent needed to give Dirichlet's original proof of his theorem on arithmetic progressions. That proof, unlike later ones, uses Dirichlet's class number formula for quadratic fields, and is worth a look.

There is a lengthy but now dated bibliography.

An unusual feature is a table (from Sommer's 1911 book) describing the structure of Z[sqrt(n)] for all nonsquare n from -99 to 99.

nice text, but somewhat hard to read4
This book gives a nice treatment of the theory of quadratic number rings and fields. It also covers a few other topics like class field theory and the Dirichlet problem. It requires surprisingly little algebra, but this makes some parts of the text seem to drag if you already know the algebra he covers. Not really recommended for independent study, since some arguments are a bit confusing, and it has few examples.

Advanced, but now dated. Still useful.4
This is definitely an advanced book. But no book claiming to be advanced can hold that title for long since mathematical research is progressive. As advanced as the book is, it's just an introduction to advanced number theory now, and dated in places. This book was orginally published as "A Second Course in Number Theory " in 1962. I own several books by Harvey Cohn and I appreciate his writing style. He writes with the complete book in mind and every chapter and paragraph is cohesively developed. His writing (between the numerous equations, tables and proofs) is lucid and conversational with historical motivation. He places a strong emphasis on ideal theory, and quadratic fields. In this regard the book is almost redundant given his "Class Field Theory" book.

Be warned there is some dated material in this book. It is prior to Alan Baker's 1966 proof about d=-163 and imaginary quadratic fields, and such is still only conjectured in the text. And of course, FLT wasn't on Wiles' check list when this book was published.

It doesn't cover prime-producing polynomials or transcendental functions and their relation to class field theory, like one would hope (I guess the world had to wait for Baker for that). And forget about rational points on elliptic curves, none at all. It's from the period when elliptic equations were poo-pooed as relics before being brought to the fore again by recent developments.

Despite all the short-comings, I can still recommend the book as a worthy edition to your number theory library. Just don't put it at the top of your lists (unless you're short on cash and Dover is all you can afford).

Linear Algebra & Its Apps 10 Volumes by Richard Brualdi

Product Details
Published on: 1997
Binding: Paperback

Elementary Linear Algebra by Howard Anton

Product Description

This classic treatment of linear algebra presents the fundamentals in the clearest possible way, examining basic ideas by means of computational examples and geometrical interpretation. It proceeds from familiar concepts to the unfamiliar, from the concrete to the abstract. Readers consistently praise this outstanding text for its expository style and clarity of presentation.
Clear, accessible, step-by-step explanations make the material crystal clear.
The authors spotlight the relationships between concepts to give a unified and complete picture.
Established the intricate thread of relationships between systems of equations, matrices, determinants, vectors, linear transformations and eigenvalues.
Product Details
Amazon Sales Rank: #44768 in Books
Published on: 2004-12-27
Number of items: 1
Binding: Hardcover
624 pages
Customer Reviews

A decent textbook
The book isn't the best organized in some regards, but it's decently written and decently eyeball-scan-friendly.

The ninth edition isn't as good as previous editions
I've been using Anton for my linear algebra classes since 1984, and have gone through nine editions. The ninth edition seems much more confusing than previous editions. He often moves text around without seeing that it makes sense in the new context: a theorem will use notation that isn't introduced until later, for example. He comes back to eigenvalues and eigenvectors over and over, looking at them from different perspectives. This should work if done well, but instead it confuses my students. I'm going to look for another book for the next time I teach the class.

No practical application....
This book mostly consists of a bunch of long droning proofs and severely lacks any real world applications. If your looking for an enjoyable learning experience, this is not the book.

A Course in Universal Algebra (Graduate texts in mathematics)

Product Details
Amazon Sales Rank: #2019747 in Books
Published on: 1982-03
Number of items: 1
Binding: Hardcover
276 pages
Customer Reviews

Though I haven't read it yet,
I found it(the millenium edition) online as a pdf or ps file. I think this book is good because I see it in many bibliography of the books of related subjects. Try any search engine under the book title.

Beautiful book. I still don't have one and I'm mad about it
The book is excellent introduction to Universal Algebra. In fact, I cannot imagine a person studying the subject who is not familiar with this (already) classic book. The only bad part is that it is impossible to find. Please, print it again. I'll by 10, two for me (one for the office and one for home) and the rest for my friends.

Very good but HARD to find!
This is one of the more recent basic texts on Universal Algebra. It is very well written, and is an absolute MUST for students of the subject, as well as a great resource for researchers. Unfortunately, it is out of print and extremely difficult to find. I have been looking for a copy in English for seven years with no success. Fortunately, there is a Hungarian edition (with a special appendix on Clones) available to those willing to break the language barrier (no small feat...) When will this classic text be reprinted?

Algebraic Statistics: Computational Commutative Algebra in Statistics (Monographs on Statistics and Applied Probability) by Giovanni Pistone

Product Description

Written by pioneers in this exciting new field, Algebraic Statistics introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. It begins with an introduction to Gröbner bases and a thorough description of their applications to experimental design. A special chapter covers the binary case with new application to coherent systems in reliability and two level factorial designs. The work paves the way, in the last two chapters, for the application of computer algebra to discrete probability and statistical modelling through the important concept of an algebraic statistical model. As the first book on the subject, Algebraic Statistics presents many opportunities for spin-off research and applications and should become a landmark work welcomed by both the statistical community and its relatives in mathematics and computer science.
Product Details
Amazon Sales Rank: #1173867 in Books
Published on: 2000-12-21
Number of items: 1
Binding: Hardcover
184 pages
Editorial Reviews

Review
...authors have been the predominant contributors to the field.... for anyone who wants to learn about, and perhaps contribute to, the field, this monograph is undoubtedly the place to start.
-Biometrics, Vol. 57, No. 3, September 2001
This very challenging monograph demonstrates how Gröbner bases may be used to represent experimental design, probability models and statistical models The book points clearly to the future potential use of algebraic tools.

Short Book Reviews, Vol. 21, No. 2, August, 2001

Book Info
An introduction to the use of polynomial algebra in experimental design, discrete probability and statistics. Presents many opportunities for spin-off research and paves the way for that application of computer algebra to probability and statistics. DLC: Mathematical statistics.

Differential Galois Theory and Non-Integrability of Hamiltonian Systems (Progress in Mathematics) by Juan J. Morales Ruiz

Product Description


This book is devoted to the relation between two different concepts of integrability: the complete integrability of complex analytical Hamiltonian systems and the integrability of complex analytical linear differential equations. For linear differential equations, integrability is made precise within the framework of differential Galois theory. The connection of these two integrability notions is given by the variational equation (i.e. linearized equation) along a particular integral curve of the Hamiltonian system. The underlying heuristic idea, which motivated the main results presented in this monograph, is that a necessary condition for the integrability of a Hamiltonian system is the integrability of the variational equation along any of its particular integral curves. This idea led to the algebraic non-integrability criteria for Hamiltonian systems. These criteria can be considered as generalizations of classical non-integrability results by Poincaré and Liapunov, as well as more recent results by Ziglin and Yoshida. Thus, by means of the differential Galois theory it is not only possible to understand all these approaches in a unified way but also to improve them. Several important applications are also included: homogeneous potentials, Bianchi IX cosmological model, three-body problem, Hénon-Heiles system, etc.

The book is based on the original joint research of the author with J.M. Peris, J.P. Ramis and C. Simó, but an effort was made to present these achievements in their logical order rather than their historical one. The necessary background on differential Galois theory and Hamiltonian systems is included, and several new problems and conjectures which open new lines of research are proposed.
Product Details
Amazon Sales Rank: #294990 in Books
Published on: 1999-09-24
Number of items: 1
Binding: Hardcover
168 pages
Editorial Reviews

Review

"...[an] account of recent work of the author and co-workers on obstructions to the complete integrability of complex Hamiltonian systems. The methods are of considerable importance to practitioners... The book provides all the needed background...and presents concrete examples in considerable detail... The final chapter...includes a fascinating account of work-in-progress by the author and his collaborators... Of particular interest...is the program of extending the differential Galois theory to higher-order variational equations... [an] excellent introduction to non-integrability methods in Hamiltonian mechanics [that] brings the reader to the forefront of research in the area. The inclusion of a large number of worked-out examples, many of wide applied interest, is commendable. There are many historical references, and an extensive bibliography."

--Mathematical Reviews

A Computational Introduction to Number Theory and Algebra by Victor Shoup

Product Description

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.
Product Details
Amazon Sales Rank: #125444 in Books
Published on: 2005-06-06
Number of items: 1
Binding: Hardcover
534 pages
Editorial Reviews

Download Description
Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

Review
"This is an outstanding and well-written book whose aim is to introduce the reader to a broad range of material -- ranging from basic to relatively advanced -- without requiring any prior knowledge on the part of the reader other than calculus and mathematical maturity. That the book succeeds at this goal is quite an accomplishment! ...this book is a must-read for anyone interested in computational number theory or algebra and especially applications of the latter to cryptography. I would not hesitate, though, to recommend this book even to students 'only' interested in the algebra itself (and not the computational aspects thereof); especially for computer science majors, this book is one of the best available introductions to that subject."
SIGACT News

“As computer science students have very likely not previously mastered probability theory, linear algebra, or basic abstract algebra, Shoup packages crash courses in each. Despite taking time for so many basics, Shoup climaxes with careful treatment of the late-breaking, ingenious, polynomial-time deterministic primality test of Agrawal, Kayal, and Saxena. Apart from number theory, one could easily build a fine discrete mathematics course on this book. Highly recommended.”
Choice

Download Description
Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.
Customer Reviews

Really a treasure
I'm a student digging into the cryptology for an year. The more article I read, the more confusion I encounter because of my poor mathematical background. However, when I get this, I could find answer to my puzzles, and make an more explicit way to settle down my own idea.

The background you really need, clear and sweet
This book is a marvel. It is clear and concise yet thorough. The author is obviously a bit of an obsessive compulsive, he has found the shortest paths from the clearest definitions to the most important results, each given with the cleanest, most insight-inducing proofs ... the results (and definitions) he gives are the ones any student (practitioner!) of modern computer science (especially cryptology) *needs* to know -- having this book on your shelves (and its contents in your head) should be a requirement for any degree, at any level, in computer science.
[Caveat: I know the author and have read his book in draft form. I also required my students to get it and read it, in a computer science course I taught.

A Computational Introduction to Number Theory and Algebra by Victor Shoup

Product Description

Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.
Product Details

* Amazon Sales Rank: #535077 in Books
* Published on: 2005-06-06
* Number of items: 1
* Binding: Hardcover
* 534 pages

Editorial Reviews

Review
"This is an outstanding and well-written book whose aim is to introduce the reader to a broad range of material -- ranging from basic to relatively advanced -- without requiring any prior knowledge on the part of the reader other than calculus and mathematical maturity. That the book succeeds at this goal is quite an accomplishment! ...this book is a must-read for anyone interested in computational number theory or algebra and especially applications of the latter to cryptography. I would not hesitate, though, to recommend this book even to students 'only' interested in the algebra itself (and not the computational aspects thereof); especially for computer science majors, this book is one of the best available introductions to that subject."
SIGACT News

“As computer science students have very likely not previously mastered probability theory, linear algebra, or basic abstract algebra, Shoup packages crash courses in each. Despite taking time for so many basics, Shoup climaxes with careful treatment of the late-breaking, ingenious, polynomial-time deterministic primality test of Agrawal, Kayal, and Saxena. Apart from number theory, one could easily build a fine discrete mathematics course on this book. Highly recommended.”
Choice

Download Description
Number theory and algebra play an increasingly significant role in computing and communications, as evidenced by the striking applications of these subjects to such fields as cryptography and coding theory. This introductory book emphasises algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The mathematical prerequisites are minimal: nothing beyond material in a typical undergraduate course in calculus is presumed, other than some experience in doing proofs - everything else is developed from scratch. Thus the book can serve several purposes. It can be used as a reference and for self-study by readers who want to learn the mathematical foundations of modern cryptography. It is also ideal as a textbook for introductory courses in number theory and algebra, especially those geared towards computer science students.

About the Author
Victor Shoup is Associate Professor at The Courant Institute of Mathematical Sciences at New York University.
Customer Reviews

Really a treasure5
I'm a student digging into the cryptology for an year. The more article I read, the more confusion I encounter because of my poor mathematical background. However, when I get this, I could find answer to my puzzles, and make an more explicit way to settle down my own idea.

The background you really need, clear and sweet5
This book is a marvel. It is clear and concise yet thorough. The author is obviously a bit of an obsessive compulsive, he has found the shortest paths from the clearest definitions to the most important results, each given with the cleanest, most insight-inducing proofs ... the results (and definitions) he gives are the ones any student (practitioner!) of modern computer science (especially cryptology) *needs* to know -- having this book on your shelves (and its contents in your head) should be a requirement for any degree, at any level, in computer science.
[Caveat: I know the author and have read his book in draft form. I also required my students to get it and read it, in a computer science course I taught.]

Invitation to Dynamical Systems by Edward R. Scheinerman

Product Details
Amazon Sales Rank: #1694887 in Books
Published on: 1995-08-04
Number of items: 1
Binding: Hardcover
384 pages
Editorial Reviews

The publisher, Prentice-Hall Engineering/Science/Mathematics
With this unique text, Scheinerman invites students from a wide range of backgrounds with limited technical prerequisites to explore the beauty and excitement of dynamical systems and of mathematics in general. The text is designed for students who want to continue exploring mathematics beyond linear algebra, but are not ready for highly abstract material. Rather than taking a dry theorem-proof-corollark-remark style, the text stresses the geometry, intuition, and appreciation of dynamical systems.

From the Back Cover
With this unique book, Scheinerman invites readers from a wide range of backgrounds with limited technical prerequisites to explore the beauty and excitement of dynamical systems in particular, and of mathematics in general. The book is designed for readers who want to continue exploring mathematics beyond linear algebra, but are not ready for highly abstract material. Rather than taking a standard mathematical theorem-proof-corollary-remark approach to dynamical systems, it stresses the intuition, ideology, and appreciation that is open to everyone.
Customer Reviews

Author's superior explanation
This book by scheinerman is a book that very well explains the thory behind higher level linear Algebra. He is a very good author, and knows how to take complex material and explain it in a simplistic way. I

Linear Algebra and Its Applications, Third Updated Edition by David C. Lay

Product Description


Linear algebra is relatively easy for students during the early stages of the course, when the material is presented in a familiar, concrete setting. But when abstract concepts are introduced, students often hit a brick wall. Instructors seem to agree that certain concepts (such as linear independence, spanning, subspace, vector space, and linear transformations), are not easily understood, and require time to assimilate. Since they are fundamental to the study of linear algebra, students' understanding of these concepts is vital to their mastery of the subject. Lay introduces these concepts early in a familiar, concrete Rn setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.
Product Details
Amazon Sales Rank: #13349 in Books
Published on: 2005-09-01
Number of items: 1
Binding: Hardcover
576 pages
Editorial Reviews

About the Author

David C. Lay holds a B.A. from Aurora University (Illinois), and an M.A. and Ph.D. from the University of California at Los Angeles. Lay has been an educator and research mathematician since 1966, mostly at the University of Maryland, College Park. He has also served as a visiting professor at the University of Amsterdam, the Free University in Amsterdam, and the University of Kaiserslautern, Germany. He has over 30 research articles published in functional analysis and linear algebra.

As a founding member of the NSF-sponsored Linear Algebra Curriculum Study Group, Lay has been a leader in the current movement to modernize the linear algebra curriculum. Lay is also co-author of several mathematics texts, including Introduction to Functional Analysis, with Angus E. Taylor, Calculus and Its Applications, with L.J. Goldstein and D.I. Schneider, and Linear Algebra Gems-Assets for Undergraduate Mathematics, with D. Carlson, C.R. Johnson, and A.D. Porter.

A top-notch educator, Professor Lay has received four university awards for teaching excellence, including, in 1996, the title of Distinguished Scholar-Teacher of the University of Maryland. In 1994, he was given one of the Mathematical Association of America's Awards for Distinguished College or Unviersity Teaching of Mathematics. He has been elected by the university students to membership in Alpha Lambda Delta National Scholastic Honor Society and Golden Key National Honor Society. In 1989, Aurora University conferred on him the Outstanding Alumnus award. Lay is a member of the American Mathematical Society, the Canadian Mathematical Society, the International Linear Algebra Society, the Mathematical Association of America, Sigma Xi, and the Society for Industrial and Applied Mathematics. Since 1992, he has served several terms on the national board of the Association of Christians in the Mathematical Sciences.
Customer Reviews

Pretty bad
This book starts out well, but eventually the chapters are too short and don't go far enough in depth on the more advanced topics. This book isn't ideal for any sort of self-study, as it does not contain enough information to thoroughly educate the reader on many of the subjects without some supplementary instruction. Also does not go into any detail on using Mat Lab or any other form of programming to solve linear algebra, which is somewhat essential nowadays. Wouldn't recommend.

An outstanding introduction to Linear Algebra
This book provides a good companion for an introductory course in Linear Algebra. Mr. Lay's style is very clear, readable, and each concept logically builds on the last. My only concern is that, like another reviewer said there is the occasional gap between the exercises and the examples presented, which may require the assistance of the instructor. 4/5 Stars

A very good introduction to linear algebra
The highest quality of a book is the ability to teach yourself from it. Lay's book is very self-teachable because it is written in a non-pretentious, explanatory way, making sure you get the big picture while making sure you can do the little details. It reminds me of Griffiths books in physics.

It is a little proof light, so I can respect that a mathematician who is into analysis might find this book too easy. Problems aren't too hard but aren't too easy for the more conceptual questions.

And I appreciate that the problems are meant to test your ability to understand the material, not do mindless calculations that I know anyone can do. For example, some matrices will just start off already diagonalized for you in later chapters.

This is written from the perspective of a physicist. I thus say to my fellow scientists that this is a great book to gain a good understanding of the linear algebra. If you are an experimentalist who frankly wants to learn only what he needs to in order to get by, THEN THIS BOOK ISN'T FOR YOU. This book develops from scratch everything you need to know for undergraduate physics. Go read a Differential Equations book and learn as you go for the linear algebra. If you're a theorist, this is for you.