xij CONTENTS l6.7 Programming Exercise·.····· 273 References and Suggested Readings.... 273 16.8 Sage 274 16.9 Sage Exercises 282 17 Polynomials 283 17.1 Polynomial Rings .. 283 17.2 The Division Algorithm 286 17.3 Irreducible Polynomials 289 17.4 Exercises 294 17.5 Additional Exercises:Solving the Cubic and Quartic Equations 296 17.6Sage...·.····· 298 17.7 Sage Exercises 303 18 Integral Domains 304 18.1 Fields of Fractions ...... 304 18.2 Factorization in Integral Domains........ 307 l8.3 Exercises················· 314 References and Suggested Readings..·· 316 18.4 Sage 316 18.5 Sage Exercises 319 19 Lattices and Boolean Algebras 320 19.1 Lattices.·· 320 19.2 Boolean Algebras......... 323 19.3 The Algebra of Electrical Circuits.... 328 19.4 Exercises··.········· 330 19.5 Programming Exercises 332 References and Suggested Readings.. 333 19.6Sage.············· 333 19.7 Sage Exercises 338 20 Vector Spaces 340 20.1 Definitions and Examples 340 20.2 Subspaces...:...·.··· 341 20.3 Linear Independence..... 342 20.4 Exercises 344 References and Suggested Readings.. 346 20.5Sage......·..····· 347 20.6 Sage Exercises 351 21 Fields 354 21.1 Extension Fields 354 21.2 Splitting Fields 362 21.3 Geometric Constructions 365 21.4 Exercises.·.······· 369 References and Suggested Readings.. 371 21.5Sage.·.················ 371 21.6 Sage Exercises........... 378xii CONTENTS 16.7 Programming Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 273 16.8 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 16.9 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 17 Polynomials 283 17.1 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 17.2 The Division Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 17.3 Irreducible Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 17.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 17.5 Additional Exercises: Solving the Cubic and Quartic Equations . . . . . . . 296 17.6 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 17.7 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 18 Integral Domains 304 18.1 Fields of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 18.2 Factorization in Integral Domains . . . . . . . . . . . . . . . . . . . . . . . . 307 18.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 316 18.4 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 18.5 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 19 Lattices and Boolean Algebras 320 19.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 19.2 Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 19.3 The Algebra of Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . 328 19.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 19.5 Programming Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 333 19.6 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 19.7 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 20 Vector Spaces 340 20.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 20.2 Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 20.3 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 20.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 346 20.5 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 20.6 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 21 Fields 354 21.1 Extension Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 21.2 Splitting Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 21.3 Geometric Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 21.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 371 21.5 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 21.6 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378