A Classical Introduction to Modern Number Theory

1. Front Matter

   Pages i-xiii

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2. Unique Factorization

   • Kenneth Ireland, Michael Rosen

   Pages 1-16

3. Applications of Unique Factorization

   • Kenneth Ireland, Michael Rosen

   Pages 17-27

4. Congruence

   • Kenneth Ireland, Michael Rosen

   Pages 28-38

5. The Structure of U(ℤ/nℤ)

   • Kenneth Ireland, Michael Rosen

   Pages 39-49

6. Quadratic Reciprocity

   • Kenneth Ireland, Michael Rosen

   Pages 50-65

7. Quadratic Gauss Sums

   • Kenneth Ireland, Michael Rosen

   Pages 66-78

8. Finite Fields

   • Kenneth Ireland, Michael Rosen

   Pages 79-87

9. Gauss and Jacobi Sums

   • Kenneth Ireland, Michael Rosen

   Pages 88-107

10. Cubic and Biquadratic Reciprocity

    • Kenneth Ireland, Michael Rosen

    Pages 108-137

11. Equations over Finite Fields

    • Kenneth Ireland, Michael Rosen

    Pages 138-150

12. The Zeta Function

    • Kenneth Ireland, Michael Rosen

    Pages 151-171

13. Algebraic Number Theory

    • Kenneth Ireland, Michael Rosen

    Pages 172-187

14. Quadratic and Cyclotomic Fields

    • Kenneth Ireland, Michael Rosen

    Pages 188-202

15. The Stickelberger Relation and the Eisenstein Reciprocity Law

    • Kenneth Ireland, Michael Rosen

    Pages 203-227

16. Bernoulli Numbers

    • Kenneth Ireland, Michael Rosen

    Pages 228-248

17. Dirichlet L-functions

    • Kenneth Ireland, Michael Rosen

    Pages 249-268

18. Diophantine Equations

    • Kenneth Ireland, Michael Rosen

    Pages 269-296

19. Elliptic Curves

    • Kenneth Ireland, Michael Rosen

    Pages 297-318

20. Back Matter

    Pages 319-344

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This book is a revised and greatly expanded version of our book Elements of Number Theory published in 1972. As with the first book the primary audience we envisage consists of upper level undergraduate mathematics majors and graduate students. We have assumed some familiarity with the material in a standard undergraduate course in abstract algebra. A large portion of Chapters 1-11 can be read even without such background with the aid of a small amount of supplementary reading. The later chapters assume some knowledge of Galois theory, and in Chapters 16 and 18 an acquaintance with the theory of complex variables is necessary. Number theory is an ancient subject and its content is vast. Any intro­ ductory book must, of necessity, make a very limited selection from the fascinat ing array of possible topics. Our focus is on topics which point in the direction of algebraic number theory and arithmetic algebraic geometry. By a careful selection of subject matter we have found it possible to exposit some rather advanced material without requiring very much in the way oftechnical background. Most of this material is classical in the sense that is was dis­ covered during the nineteenth century and earlier, but it is also modern because it is intimately related to important research going on at the present time.

• Zahlentheorie
• algebra
• algebraic number theory
• arithmetic
• diophantine equation
• elliptic curve
• finite field
• geometry
• number theory
• zeta function

From the reviews of the second edition:

K. Ireland and M. Rosen

A Classical Introduction to Modern Number Theory

"Many mathematicians of this generation have reached the frontiers of research without having a good sense of the history of their subject. In number theory this historical ignorance is being alleviated by a number of fine recent books. This work stands among them as a unique and valuable contribution."

— MATHEMATICAL REVIEWS

"This is a great book, one that does exactly what it proposes to do, and does it well. For me, this is the go-to book whenever a student wants to do an advanced independent study project in number theory. … for a student who wants to get started on the subject and has taken a basic course on elementary number theory and the standard abstract algebra course, this is perfect." (Fernando Q. Gouvêa, MathDL, January, 2006)

• Department of Mathematics, University of New Brunswick, Fredericton, Canada

  Kenneth Ireland

• Department of Mathematics, Brown University, Providence, USA

  Michael Rosen

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