## A Classical Introduction to Modern Number TheoryBridging the gap between elementary number theory and the systematic study of advanced topics, A Classical Introduction to Modern Number Theory is a well-developed and accessible text that requires only a familiarity with basic abstract algebra. Historical development is stressed throughout, along with wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. An extensive bibliography and many challenging exercises are also included. This second edition has been corrected and contains two new chapters which provide a complete proof of the Mordell-Weil theorem for elliptic curves over the rational numbers, and an overview of recent progress on the arithmetic of elliptic curves. |

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#### Review: A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (Graduate Texts in Mathematics #84)

User Review - Dan - GoodreadsI just read the chapter about the elliptic curve y^2 = x^3 + Dx, it was pretty good. Read full review

#### Review: A Classical Introduction to Modern Number Theory (Graduate Texts in Mathematics) (Graduate Texts in Mathematics #84)

User Review - Arman - GoodreadsThe most difficult book I have read up to now ;).For me it took more than an hour to read one page of this book ! This book devotes some parts about historical comments in number theory, have you ever ... Read full review

### Contents

Chapter 1 | 1 |

2 Unique Factorization in X | 6 |

3 Unique Factorization in a Principal Ideal Domain | 8 |

4 The Rings Zi and Zw | 12 |

Chapter 2 | 17 |

2 Some Arithmetic Functions | 18 |

3 1p Diverges | 21 |

4 The Growth of nx | 22 |

3 Ramification and Degree | 181 |

Chapter 13 | 188 |

2 Cyclotomic Fields | 193 |

3 Quadratic Reciprocity Revisited | 199 |

2 The Power Residue Symbol | 204 |

3 The Stickelberger Relation | 207 |

4 The Proof of the Stickelberger Relation | 209 |

5 The Proof of the Eisenstein Reciprocity Law | 215 |

Chapter 3 | 28 |

2 Congruence in Z | 29 |

3 The Congruence ax b m | 31 |

4 The Chinese Remainder Theorem | 34 |

Chapter 4 | 39 |

2 nth Power Residues | 45 |

Chapter 5 | 50 |

2 Law of Quadratic Reciprocity | 53 |

3 A Proof of the Law of Quadratic Reciprocity | 58 |

Chapter 6 | 66 |

2 The Quadratic Character of 2 | 69 |

3 Quadratic Gauss Sums | 70 |

4 The Sign of the Quadratic Gauss Sum | 73 |

Chapter 7 | 79 |

2 The Existence of Finite Fields | 83 |

3 An Application to Quadratic Residues | 85 |

Chapter 8 | 88 |

2 Gauss Sums | 91 |

3 Jacobi Sums | 92 |

4 The Equation x + y 1 in F | 97 |

5 More on Jacobi Sums | 98 |

6 Applications | 101 |

7 A General Theorem | 102 |

Chapter 9 | 108 |

1 The Ring Zo | 109 |

2 Residue Class Rings | 111 |

3 Cubic Residue Character | 112 |

4 Proof of the Law of Cubic Reciprocity | 115 |

5 Another Proof of the Law of Cubic Reciprocity | 117 |

6 The Cubic Character of 2 | 118 |

Preliminaries | 119 |

8 The Quartic Residue Symbol | 121 |

9 The Law of Biquadratic Reciprocity | 123 |

10 Rational Biquadratic Reciprocity | 127 |

11 The Constructibility of Regular Polygons | 130 |

12 Cubic Gauss Sums and the Problem of Kummer | 131 |

Chapter 10 | 138 |

2 Chevalleys Theorem | 143 |

3 Gauss and Jacobi Sums over Finite Fields | 145 |

Chapter 11 | 151 |

2 Trace and Norm in Finite Fields | 158 |

3 The Rationality of the Zeta Function Associated to a0x + tfjxT + + attx | 161 |

4 A Proof of the HasseDavenport Relation | 163 |

5 The Last Entry | 166 |

Chapter 12 | 172 |

2 Unique Factorization in Algebraic Number Fields | 174 |

6 Three Applications | 220 |

Chapter 15 | 228 |

2 Congruences Involving Bernoulli Numbers | 234 |

3 Herbrands Theorem | 241 |

Chapter 16 | 249 |

2 A Special Case | 251 |

3 Dirichlet Characters | 253 |

4 Dirichlet Lfunctions | 255 |

5 The Key Step | 257 |

6 Evaluating Ls at Negative Integers | 261 |

Chapter 17 | 269 |

2 The Method of Descent | 271 |

3 Legendres Theorem | 272 |

3 Sophie Germains Theorem | 275 |

5 Pells Equation | 276 |

6 Sums of Two Squares | 278 |

7 Sums of Four Squares | 280 |

Exponent 3 | 284 |

9 Cubic Curves with Infinitely Many Rational Points | 287 |

10 The Equation2 x3 + k | 288 |

11 The First Case of Fermats Conjecture for Regular Exponent | 290 |

12 Diophantine Equations and Diophantine Approximation | 292 |

Chapter 18 | 297 |

2 Local and Global Zeta Functions of an Elliptic Curve | 301 |

3 y2 x3 + D the Local Case | 304 |

4 M y2 x3 Dx the Local Case | 306 |

5 Hecke Lfunctions | 307 |

6 y x Dx the Global Case | 310 |

7 y2 x3 + D the Global Case | 312 |

8 Final Remarks | 314 |

Chapter 19 | 319 |

2 The Group E2E | 323 |

3 The Weak Dirichlet Unit Theorem | 326 |

4 The Weak MordellWeil Theorem | 328 |

5 The Descent Argument | 329 |

Chapter 20 | 339 |

1 The Mordell Conjecture | 340 |

2 Elliptic Curves | 343 |

3 Modular Curves | 345 |

4 Heights and the Height Regulator | 347 |

5 New Results on the BirchSwinnertonDyer Conjecture | 353 |

6 Applications to Gausss Class Number Conjecture | 358 |

Selected Hints for the Exercises | 367 |

Bibliography | 375 |

385 | |

### Common terms and phrases

### References to this book

### References from web pages

JSTOR: A Classical Introduction to Modern Number Theory

A classical introduction to modern number theory (2nd edition), by Kenneth Ireland and Michael Rosen. Pp 394. DM 98. 1990. ISBN 3-540-97329-X (Springer) ...

links.jstor.org/ sici?sici=0025-5572(199207)2%3A76%3A476%3C316%3AACITMN%3E2.0.CO%3B2-W

planetmath: bibliography for number theory

Daniel A. Marcus, Number Fields, Springer, New York. K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, 1998. ...

planetmath.org/ encyclopedia/ BibliographyForNumberTheory.html

Number Theory - Mathematics 156

Text, A Classical Introduction to Modern Number Theory (2nd edition) by Kenneth Ireland and Michael Rosen Springer-Verlag GTM Series, ISBN 038797329X ...

www.math.brown.edu/ ~jhs/ MA0156/ MA0156HomePage.html

Selected Number Theory References

($95, UMBC library); [IR91] A Classical Introduction to Modern Number Theory, 2nd Ed, by Ireland & Rosen, Springer-Verlag, 1991: One of the best general ...

www.math.umbc.edu/ ~campbell/ NumbThy/ Class/ References.html

MAT 311 - Number Theory -- Spring 2004

... An Introduction to the Theory of Numbers, I. Niven and hs Zuckerman; A Classical Introduction to Modern Number Theory, K. Ireland and M. Rosen ...

www.math.sunysb.edu/ ~sorin/ 311-2004/

Zentralblatt MATH Database 1931 – 2008 1026.11001

M. Rosen [A classical introduction to modern number theory (Graduate Texts in Math-. ematics 84, Springer, New York) (1982; Zbl 0482.10001)]. ...

zmath.impa.br/ cgi-bin/ zmen/ ZMATH/ en/ quick.html?first=1&

Department of Mathematics BOOK LIST FOR 2007-2008 012a

A Classical Introduction to Modern Number Theory, Second Edition by K. Ireland. and M. Rosen, published by Springer Verlag. (Optional) ...

www.math.uwo.ca/ undergrad/ Booklist071009.pdf

Kaneko: Poly-Bernoulli numbers

[2] Ireland, K. and Rosen, M.: A Classical Introduction to Modern Number Theory, second edition. Springer GTM 84 (1990) [3] Jordan, Charles: Calculus of ...

jtnb.cedram.org/ jtnb-bin/ fitem?id=JTNB_1997__9_1_221_0

Literatura iz teorije brojeva

K. Ireland, M. Rosen: A Classical Introduction to Modern Number Theory, Springer-Verlag, 1998. (B,D). ga Jones, jm Jones: Elementary Number Theory, ...

web.math.hr/ ~duje/ literatura.html

Foreign Dispatches: A History of Dandyism

kf Ireland and M. Rosen: A Classical Introduction to Modern Number Theory · kf Ireland and M. Rosen: A Classical Introduction to Modern Number Theory ...

foreigndispatches.typepad.com/ dispatches/ 2005/ 02/ a_history_of_da.html