Introductory Algebraic Number Theory

by Saban Alaca , Kenneth S. Williams

• ISBN13: 9780521832502
• ISBN10: 0521832500
• Format: Hardcover
• Copyright: 2003-11-17
• Publisher: Cambridge University Press

Summary

Algebraic number theory is a subject which came into being through the attempts of mathematicians to try to prove Fermat's last theorem and which now has a wealth of applications to diophantine equations, cryptography, factoring, primality testing and public-key cryptosystems. This book provides an introduction to the subject suitable for senior undergraduates and beginning graduate students in mathematics. The material is presented in a straightforward, clear and elementary fashion, and the approach is hands on, with an explicit computational flavour. Prerequisites are kept to a minimum, and numerous examples illustrating the material occur throughout the text. References to suggested reading and to the biographies of mathematicians who have contributed to the development of algebraic number theory are given at the end of each chapter. There are over 320 exercises, an extensive index, and helpful location guides to theorems and lemmas in the text.

Table of Contents

List of Tables	xi
Notation	xiii
Introduction	xv
1 Integral Domains	1	(26)
1.1 Integral Domains	1	(4)
1.2 Irreducibles and Primes	5	(3)
1.3 Ideals	8	(2)
1.4 Principal Ideal Domains	10	(6)
1.5 Maximal Ideals and Prime Ideals	16	(5)
1.6 Sums and Products of Ideals	21	(2)
Exercises	23	(2)
Suggested Reading	25	(1)
Biographies	25	(2)
2 Euclidean Domains	27	(27)
2.1 Euclidean Domains	27	(3)
2.2 Examples of Euclidean Domains	30	(7)
2.3 Examples of Domains That are Not Euclidean	37	(9)
2.4 Almost Euclidean Domains	46	(1)
2.5 Representing Primes by Binary Quadratic Forms	47	(2)
Exercises	49	(2)
Suggested Reading	51	(2)
Biographies	53	(1)
3 Noetherian Domains	54	(20)
3.1 Noetherian Domains	54	(3)
3.2 Factorization Domains	57	(3)
3.3 Unique Factorization Domains	60	(4)
3.4 Modules	64	(3)
3.5 Noetherian Modules	67	(4)
Exercises	71	(1)
Suggested Reading	72	(1)
Biographies	73	(1)
4 Elements Integral over a Domain	74	(14)
4.1 Elements Integral over a Domain	74	(7)
4.2 Integral Closure	81	(5)
Exercises	86	(1)
Suggested Reading	87	(1)
Biographies	87	(1)
5 Algebraic Extensions of a Field	88	(21)
5.1 Minimal Polynomial of an Element Algebraic over a Field	88	(2)
5.2 Conjugates of α over Κ	90	(1)
5.3 Conjugates of an Algebraic Integer	91	(3)
5.4 Algebraic Integers in a Quadratic Field	94	(4)
5.5 Simple Extensions	98	(4)
5.6 Multiple Extensions	102	(4)
Exercises	106	(2)
Suggested Reading	108	(1)
Biographies	108	(1)
6 Algebraic Number Fields	109	(32)
6.1 Algebraic Number Fields	109	(3)
6.2 Conjugate Fields of an Algebraic Number Field	112	(4)
6.3 The Field Polynomial of an Element of an Algebraic Number Field	116	(7)
6.4 The Discriminant of a set of Elements in an Algebraic Number Field	123	(6)
6.5 Basis of an Ideal	129	(8)
6.6 Prime Ideals in Rings of Integers	137	(1)
Exercises	138	(2)
Suggested Reading	140	(1)
Biographies	140	(1)
7 Integral Bases	141	(53)
7.1 Integral Basis of an Algebraic Number Field	141	(19)
7.2 Minimal Integers	160	(10)
7.3 Some Integral Bases in Cubic Fields	170	(8)
7.4 Index and Minimal Index of an Algebraic Number Field	178	(8)
7.5 Integral Basis of a Cyclotomic Field	186	(3)
Exercises	189	(2)
Suggested Reading	191	(2)
Biographies	193	(1)
8 Dedekind Domains	194	(24)
8.1 Dedekind Domains	194	(1)
8.2 Ideals in a Dedekind Domain	195	(5)
8.3 Factorization into Prime Ideals	200	(6)
8.4 Order of an Ideal with Respect to a Prime Ideal	206	(9)
8.5 Generators of Ideals in a Dedekind Domain	215	(1)
Exercises	216	(1)
Suggested Reading	217	(1)
9 Norms of Ideals	218	(18)
9.1 Norm of an Integral Ideal	218	(4)
9.2 Norm and Trace of an Element	222	(6)
9.3 Norm of a Product of Ideals	228	(3)
9.4 Norm of a Fractional Ideal	231	(2)
Exercises	233	(1)
Suggested Reading	234	(1)
Biographies	235	(1)
10 Factoring Primes in a Number Field	236	(28)
10.1 Norm of a Prime Ideal	236	(5)
10.2 Factoring Primes in a Quadratic Field	241	(8)
10.3 Factoring Primes in a Monogenic Number Field	249	(4)
10.4 Some Factorizations in Cubic Fields	253	(4)
10.5 Factoring Primes in an Arbitrary Number Field	257	(3)
10.6 Factoring Primes in a Cyclotomic Field	260	(1)
Exercises	261	(1)
Suggested Reading	262	(2)
11 Units in Real Quadratic Fields	264	(35)
11.1 The Units of Z+Z/2	264	(3)
11.2 The Equation x2-y2=1	267	(4)
11.3 Units of Norm 1	271	(4)
11.4 Units of Norm -1	275	(3)
11.5 The Fundamental Unit	278	(8)
11.6 Calculating the Fundamental Unit	286	(8)
11.7 The Equation x2-my2=N	294	(3)
Exercises	297	(1)
Suggested Reading	298	(1)
Biographies	298	(1)
12 The Ideal Class Group	299	(45)
12.1 Ideal Class Group	299	(1)
12.2 Minkowski's Translate Theorem	300	(5)
12.3 Minkowski's Convex Body Theorem	305	(1)
12.4 Minkowski's Linear Forms Theorem	306	(5)
12.5 Finiteness of the Ideal Class Group	311	(3)
12.6 Algorithm to Determine the Ideal Class Group	314	(17)
12.7 Applications to Binary Quadratic Forms	331	(10)
Exercises	341	(2)
Suggested Reading	343	(1)
Biographies	343	(1)
13 Dirichlet's Unit Theorem	344	(41)
13.1 Valuations of an Element of a Number Field	344	(2)
13.2 Properties of Valuations	346	(13)
13.3 Proof of Dirichlet's Unit Theorem	359	(2)
13.4 Fundamental System of Units	361	(2)
13.5 Roots of Unity	363	(6)
13.6 Fundamental Units in Cubic Fields	369	(9)
13.7 Regulator	378	(4)
Exercises	382	(1)
Suggested Reading	383	(1)
Biographies	384	(1)
14 Applications to Diophantine Equations	385	(28)
14.1 Insolvability of y2=x3+k Using Congruence Considerations	385	(4)
14.2 Solving y2=x3+k Using Algebraic Numbers	389	(12)
14.3 The Diophantine Equation y(y+1)=x(x+1)(x+2)	401	(9)
Exercises	410	(1)
Suggested Reading	411	(1)
Biographies	411	(2)
List of Definitions	413	(4)
Location of Theorems	417	(4)
Location of Lemmas	421	(2)
Bibliography	423	(2)
Index	425