Notes on Galois Theory
نویسنده
چکیده
§1. Algebraic Extensions 2 §1.1. Field extensions 2 §1.2. Multiplicativity of degree 4 §1.3. Algebraic extensions 4 §1.4. Adjoining roots 5 §1.5. Splitting fields 5 §1.6. Algebraic closure 7 §1.7. Finite fields 9 §1.8. Composite field 9 §1.9. Exercises 10 §2. Galois Theory 11 §2.1. Separable extensions 11 §2.2. Normal extensions 13 §2.3. Main Theorem of Galois Theory 13 §2.4. Fields of invariants 16 §2.5. Exercises 17 §3. Applications of Galois Theory 17 §3.1. Intermediate subfields in separable extensions 17 §3.2. Fundamental Theorem of Algebra 18 §3.3. Quadratic extensions 18 §3.4. Cubic extensions 18 §3.5. Galois group of a finite field 19 §3.6. Exercises 20 §4. Roots of unity 21 §4.1. Adjoining roots of unity 21 §4.2. Cyclotomic fields 22 §4.3. Rational cosines 23 §4.4. Abelian extensions of Q 23 §4.5. Quadratic reciprocity 25 §4.6. Exercises 26 §5. Integral extensions of rings 28 §5.1. Integral extensions 28 §5.2. Examples of the integral closure 30 §5.3. Exercises 32 §6. Cyclic extensions 32 §6.1. Cyclic extensions 32 §6.2. Artin’s Lemma 34 §6.3. Lagrange resolvents 35
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