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)

单词	TopicsOnFieldExtensionsAndGaloisTheory
释义	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)
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