A Concrete Introduction to Higher Algebra

Front Matter....Pages i-xiv
Front Matter....Pages 1-1
Numbers....Pages 3-6
Induction; the Binomial Theorem....Pages 7-18
Unique Factorization into Products of Primes....Pages 19-30
Primes....Pages 31-36
Bases....Pages 37-46
Congruences....Pages 47-55
Congruence Classes....Pages 56-61
Rings and Fields....Pages 62-67
Matrices and Vectors....Pages 68-83
Secret Codes, I....Pages 84-89
Fermat’s Theorem, I: Abelian Groups....Pages 90-100
Repeating Decimals, I....Pages 101-104
Error-correcting Codes, I....Pages 105-111
The Chinese Remainder Theorem....Pages 112-117
Secret Codes, II....Pages 118-121
Front Matter....Pages 123-123
Polynomials....Pages 125-128
Unique Factorization....Pages 129-135
The Fundamental Theorem of Algebra....Pages 136-141
Irreducible Polynomials in ℝ[ x ]....Pages 142-143
Partial Fractions....Pages 144-156
Front Matter....Pages 123-123
The Derivative of a Polynomial....Pages 157-159
Sturm’s Algorithm....Pages 160-165
Factoring in ℚ[ x ], I....Pages 166-172
Congruences Modulo a Polynomial....Pages 173-174
Fermat’s Theorem, II....Pages 175-179
Factoring in ℚ[ x ], II: Lagrange Interpolation....Pages 180-184
Factoring in ℤ p [ x ]....Pages 185-192
Factoring in ℚ[ x ], III: mod M ....Pages 193-204
Front Matter....Pages 205-205
Primitive Elements....Pages 207-211
Repeating Decimals, II....Pages 212-217
Testing for Primeness....Pages 218-221
Fourth Roots of One in ℤ p ....Pages 222-224
Telephone Cable Splicing....Pages 225-228
Factoring in ℚ[ x ], IV: Bad Examples mod p ....Pages 229-230
Congruence Classes Modulo a Polynomial: Simple Field Extensions....Pages 231-236
Polynomials and Roots....Pages 237-241
Error-Correcting Codes, II....Pages 242-254
Isomorphisms, I....Pages 255-263
Finite Fields are Simple....Pages 264-266
Latin Squares....Pages 267-272
Front Matter....Pages 205-205
Irreducible Polynomials in ℤ p [ x ]....Pages 273-279
Finite Fields....Pages 280-281
The Discriminant and Stickelberger’s Theorem....Pages 282-290
Quadratic Residues....Pages 291-298
Duplicate Bridge Tournaments....Pages 299-308
Algebraic Number Fields....Pages 309-313
Isomorphisms, II....Pages 314-315
Sums of Two Squares....Pages 316-319
On Unique Factorization....Pages 320-322
Back Matter....Pages 323-338