9 Fields

Section 9.1: Introduction
Section 9.2: Basic definitions
- Definition 9.2.1
- Definition 9.2.2
Section 9.3: Examples of fields
- Example 9.3.1: Rational numbers
- Example 9.3.2: Prime fields
- Example 9.3.3
- Example 9.3.4: Quotient fields
- Example 9.3.5: Field of rational functions
- Example 9.3.6
Section 9.4: Vector spaces
- Lemma 9.4.1
- Lemma 9.4.2
Section 9.5: The characteristic of a field
- Definition 9.5.1
- Example 9.5.2
Section 9.6: Field extensions
- Lemma 9.6.1
- Definition 9.6.2
- Definition 9.6.3
- Example 9.6.4
- Example 9.6.5
- Definition 9.6.6
- Exercise 9.6.7
- Lemma 9.6.8: Classification of simple extensions
Section 9.7: Finite extensions
- Definition 9.7.1
- Example 9.7.2
- Lemma 9.7.3
- Example 9.7.4: Degree of a rational function field
- Lemma 9.7.5
- Example 9.7.6: Degree of a simple algebraic extension
  - Equation 9.7.6.1
- Lemma 9.7.7: Multiplicativity
- Definition 9.7.8
Section 9.8: Algebraic extensions
- Definition 9.8.1
- Example 9.8.2
- Example 9.8.3
- Lemma 9.8.4
- Lemma 9.8.5
- Lemma 9.8.6 slogan
- Lemma 9.8.7
- Lemma 9.8.8
- Lemma 9.8.9
- Lemma 9.8.10
- Lemma 9.8.11
Section 9.9: Minimal polynomials
- Definition 9.9.1
- Lemma 9.9.2
Section 9.10: Algebraic closure
- Definition 9.10.1
- Lemma 9.10.2
- Definition 9.10.3
- Theorem 9.10.4
- Lemma 9.10.5
- Lemma 9.10.6
Section 9.11: Relatively prime polynomials
- Definition 9.11.1
- Lemma 9.11.2
Section 9.12: Separable extensions
- Lemma 9.12.1
- Definition 9.12.2
- Lemma 9.12.3
- Lemma 9.12.4
- Lemma 9.12.5
- Definition 9.12.6
- Situation 9.12.7
- Lemma 9.12.8
- Lemma 9.12.9
- Lemma 9.12.10
- Lemma 9.12.11
- Lemma 9.12.12
- Lemma 9.12.13
Section 9.13: Linear independence of characters
- Lemma 9.13.1
- Lemma 9.13.2
- Lemma 9.13.3
- Lemma 9.13.4
Section 9.14: Purely inseparable extensions
- Definition 9.14.1
- Lemma 9.14.2
- Lemma 9.14.3
- Lemma 9.14.4
- Lemma 9.14.5
- Lemma 9.14.6 slogan
- Definition 9.14.7
- Lemma 9.14.8
- Lemma 9.14.9: Multiplicativity
Section 9.15: Normal extensions
- Definition 9.15.1
- Lemma 9.15.2
- Lemma 9.15.3
- Lemma 9.15.4
- Lemma 9.15.5
- Lemma 9.15.6
- Lemma 9.15.7
- Definition 9.15.8
- Lemma 9.15.9
- Lemma 9.15.10
Section 9.16: Splitting fields
- Lemma 9.16.1
- Definition 9.16.2
- Lemma 9.16.3 slogan
- Definition 9.16.4
- Lemma 9.16.5
- Lemma 9.16.6
Section 9.17: Roots of unity
- Lemma 9.17.1
Section 9.18: Finite fields
Section 9.19: Primitive elements
- Lemma 9.19.1: Primitive element
Section 9.20: Trace and norm
- Definition 9.20.1
- Lemma 9.20.2
- Lemma 9.20.3
- Lemma 9.20.4
- Lemma 9.20.5
- Definition 9.20.6
- Lemma 9.20.7
- Definition 9.20.8
- Exercise 9.20.9
Section 9.21: Galois theory
- Definition 9.21.1
- Lemma 9.21.2
- Definition 9.21.3
- Lemma 9.21.4
- Lemma 9.21.5
- Lemma 9.21.6
- Theorem 9.21.7: Fundamental theorem of Galois theory
- Lemma 9.21.8
Section 9.22: Infinite Galois theory
- Lemma 9.22.1
- Lemma 9.22.2
- Lemma 9.22.3
- Theorem 9.22.4: Fundamental theorem of infinite Galois theory
- Lemma 9.22.5
Section 9.23: The complex numbers
- Lemma 9.23.1: Fundamental theorem of algebra
Section 9.24: Kummer extensions
- Lemma 9.24.1: Kummer extensions
- Lemma 9.24.2
- Lemma 9.24.3
Section 9.25: Artin-Schreier extensions
- Lemma 9.25.1: Artin-Schreier extensions
Section 9.26: Transcendence
- Definition 9.26.1
- Example 9.26.2
- Lemma 9.26.3
- Definition 9.26.4
- Lemma 9.26.5
- Example 9.26.6
- Example 9.26.7
- Example 9.26.8
- Definition 9.26.9
- Lemma 9.26.10
- Lemma 9.26.11
Section 9.27: Linearly disjoint extensions
- Definition 9.27.1
  - Equation 9.27.1.1
- Definition 9.27.2
- Lemma 9.27.3
Section 9.28: Review
- Definition 9.28.1
- Lemma 9.28.2