This upper-level undergraduate textbook provides a modern view of algebra with an eye to new applications that have arisen in recent years. A rigorous introduction to basic number theory, rings, fields, polynomial theory, groups, algebraic geometry and elliptic curves prepares students for exploring their practical applications related to storing, securing, retrieving and communicating information in the electronic world. It will serve as a textbook for an undergraduate course in algebra with a strong emphasis on applications. The book offers a brief introduction to elementary number theory as well as a fairly complete discussion of major algebraic systems (such as rings, fields, and groups) with a view of their use in bar coding, public key cryptosystems, error-correcting codes, counting techniques, and elliptic key cryptography. This is the only entry level text for algebraic systems that includes an extensive introduction to elliptic curves, a topic that has leaped to prominence due to its importance in the solution of Fermat's Last Theorem and its incorporation into the rapidly expanding applications of elliptic curve cryptography in smart cards. Computer science students will appreciate the strong emphasis on the theory of polynomials, algebraic geometry and Groebner bases. The combination of a rigorous introduction to abstract algebra with a thorough coverage of its applications makes this book truly unique.
Author(s): Norman R Reilly
Publisher: Oxford University Press
Year: 2010
Language: English
Pages: 524
City: New York
Tags: Математика;Общая алгебра;
Contents......Page 12
1.1 Sets, functions, numbers......Page 18
1.2 Induction......Page 29
1.3 Divisibility......Page 34
1.4 Prime Numbers......Page 41
1.5 Relations and Partitions......Page 47
1.6 Modular Arithmetic......Page 51
1.7 Equations in Z[sub(n)]......Page 59
1.8 Bar codes......Page 66
1.9 The Chinese Remainder Theorem......Page 71
1.10 Euler’s function......Page 76
1.11 Theorems of Euler and Fermat......Page 79
1.12 Public Key Cryptosystems......Page 83
2.1 Basic Properties......Page 94
2.2 Subrings and Subfields......Page 101
2.3 Review of Vector Spaces......Page 110
2.4 Polynomials......Page 115
2.5 Polynomial Evaluation and Interpolation......Page 124
2.6 Irreducible Polynomials......Page 131
2.7 Construction of Fields......Page 138
2.8 Extension Fields......Page 145
2.9 Multiplicative Structure of Finite Fields......Page 155
2.10 Primitive Elements......Page 158
2.11 Subfield Structure of Finite Fields......Page 163
2.12 Minimal Polynomials......Page 167
2.13 Isomorphisms between Fields......Page 176
2.14 Error Correcting Codes......Page 183
3.1 Basic Properties......Page 196
3.2 Subgroups......Page 208
3.3 Permutation Groups......Page 215
3.4 Matrix Groups......Page 222
3.5 Even and Odd Permutations......Page 228
3.6 Cayley’s Theorem......Page 232
3.7 Lagrange’s Theorem......Page 235
3.8 Orbits......Page 243
3.9 Orbit/Stabilizer Theorem......Page 249
3.10 The Cauchy-Frobenius Theorem......Page 256
3.11 K-Colorings......Page 265
3.12 Cycle Index and Enumeration......Page 269
4.1 Homomorphisms......Page 279
4.2 The Isomorphism Theorems......Page 289
4.3 Direct Products......Page 293
4.4 Finite Abelian Groups......Page 298
4.5 Conjugacy and the Class Equation......Page 307
4.6 The Sylow Theorems 1 and 2......Page 314
4.7 Sylow’s Third Theorem......Page 320
4.8 Solvable Groups......Page 324
4.9 Nilpotent Groups......Page 330
4.10 The Enigma Encryption Machine......Page 337
5.1 Homomorphisms and Ideals......Page 346
5.2 Polynomial Rings......Page 354
5.3 Division Algorithm in F[x[sub(1)], x[sub(2)], . . . , x[sub(n)]] : Single Divisor......Page 364
5.4 Multiple Divisors: Groebner Bases......Page 373
5.5 Ideals and Affine Varieties......Page 382
5.6 Decomposition of Affine Varieties......Page 391
5.7 Cubic Equations in One Variable......Page 396
5.8 Parameters......Page 398
5.9 Intersection Multiplicities......Page 409
5.10 Singular and Nonsingular Points......Page 414
6.1 Elliptic Curves......Page 418
6.2 Homogeneous Polynomials......Page 423
6.3 Projective Space......Page 429
6.4 Intersection of Lines and Curves......Page 442
6.5 Defining Curves by Points......Page 448
6.6 Classification of Conics......Page 454
6.7 Reducible Conics and Cubics......Page 458
6.8 The Nine-Point Theorem......Page 462
6.9 Groups on Elliptic Curves......Page 468
6.10 The Arithmetic on an Elliptic Curve......Page 473
6.11 Results Concerning the Structure of Groups on Elliptic Curves......Page 481
7.1 Elliptic Curve Cryptosystems......Page 486
7.2 Fermat’s Last Theorem......Page 490
7.3 Elliptic Curve Factoring Algorithm......Page 495
7.4 Singular Curves of Form y[sup(2)] = x[sup(3)] + ax + b......Page 500
7.5 Birational Equivalence......Page 503
7.6 The Genus of a Curve......Page 509
7.7 Pell’s Equation......Page 511
References......Page 518
A......Page 520
F......Page 521
M......Page 522
S......Page 523
Z......Page 524
