topics on field extensions and Galois theory

Definitions

1. extension field

2. simple field extension

3. degree (http://planetmath.org/Degree)

4. field adjunction

5. composite field

6. finite extension

7. algebraic extension

8. algebraic closure

9. abelian number field

10. abelian extension

11. biquadratic field

12. biquadratic extension

13. quadratic closure

14. cyclic extension

15. pure cubic field

16. number field

17. algebraically solvable

18. norm and trace of algebraic number

19. perfect field

20. cyclotomic field

21. cyclotomic extension

22. $p$-extension (http://planetmath.org/PExtension)

23. radical

24. radical extension

25. solvable by radicals

26. expressible

27. normal extension

28. normal closure

29. separable extension

30. separable closure

31. splitting field

32. fixed field of a set of automorphisms

33. conjugate fields

34. Galois extension

35. Galois closure

36. Galois group

37. absolute Galois group

38. Galois connection

39. inverse Galois problem

40. function field

41. ray class field

Field theory in relation to Euclidean geometry

1. Euclidean field

2. constructible numbers

3. motivation of definition of constructible numbers

4. compass and straightedge construction

5. theorem on constructible numbers

6. theorem on constructible angles

7. criterion for constructibility of regular polygon

8. classical problems of constructibility

Galois theory

1. fundamental theorem of Galois theory

2. Galois group of the compositum of two Galois extensions

3. Galois subfields of real radical extensions (http://planetmath.org/GaloisSubfieldsOfRealRootExtensionsAreAtMostQuadratic)

4. compositum of a Galois extension and another extension (http://planetmath.org/CompositumOfAGaloisExtensionAndAnotherExtensionIsGalois)

5. criterion for solvability of a polynomial by radicals (http://planetmath.org/GaloisCriterionForSolvabilityOfAPolynomialByRadicals)

6. Hilbert Theorem 90

7. Shafarevich’s theorem (finite solvable groups as Galois groups)

8. number fields with a particular finite abelian Galois group (http://planetmath.org/GaloisGroupsOfFiniteAbelianExtensionsOfMathbbQ)

Applications of Galois theory

1. Galois group of a biquadratic extension

2. cubic formula

3. quartic formula

4. Galois group of a cubic polynomial

5. Galois group of a quartic polynomial

6. Galois-theoretic derivation of the cubic formula

7. proof of fundamental theorem of algebra

8. casus irreducibilis

9. Cardano’s formulae

10. a condition of algebraic extension

11. primitive element theorem (Steinitz)