CONTENTS xiii 22 Finite Fields 380 22.1 Structure of a Finite Field....·...··..··. 380 22.2 Polynomial Codes 384 22.3 Exercises 390 22.4 Additional Exercises:Error Correction for BCH Codes 392 References and Suggested Readings.········· 393 22.5 Sage 393 22.6 Sage Exercises 395 23 Galois Theory 397 23.1 Field Automorphisms 397 23.2 The Fundamental Theorem 401 23.3 Applications.....·...··. 407 23.4 Exercises 411 References and Suggested Readings 413 23.5 Sage 414 23.6 Sage Exercises 424 A GNU Free Documentation License 427 B Hints and Solutions to Selected Exercises 434 C Notation 448 Index 451CONTENTS xiii 22 Finite Fields 380 22.1 Structure of a Finite Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 22.2 Polynomial Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 22.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 22.4 Additional Exercises: Error Correction for BCH Codes . . . . . . . . . . . . 392 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 393 22.5 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 22.6 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 23 Galois Theory 397 23.1 Field Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 23.2 The Fundamental Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 23.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 23.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 References and Suggested Readings . . . . . . . . . . . . . . . . . . . . . . . . . . 413 23.5 Sage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 23.6 Sage Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 A GNU Free Documentation License 427 B Hints and Solutions to Selected Exercises 434 C Notation 448 Index 451