Galois Theory

27 flashcards · Other · by ceyhun elmacıoğlu

1. Eisenstein’s Irreducibility Criterion

2. Rational Root Test

3. Simple Extension

4. Algebraic and Transcendental

5. Minimal Polynomial

8. True/False:

9. Algebraic Number and Field Extension

10. True/False: Automorphism permutes roots of a polynomial.

    True.

11. What is the order of the group of automorphisms of L over K?

12. Artin's Theorem

13. Galois Extension

14. What is the order of Galois group?

15. Normal Extension

16. Splitting Field

17. Are splitting fields unique?

    Yes, up to isomorphism.

18. "Normal - Splitting Field" equivalence

    Normal extension ⟺ \iff⟺ Splitting field of a polynomial

19. Separable

20. "Galois Extension" equivalence

    Galois ⟺ \iff⟺ Normal and Separable ⟺ \iff⟺ Splitting field without multiple roots

21. F∗F^\astF∗ = subgroup fixing FFF H†H^\daggerH† = subfield fixed by HHH

22. †\dagger† and ∗\ast∗

23. Galois Correspondance #2

24. Galois Correspondance #3

25. Reversing Hasse Diagrams

26. Classification of Finite Abelian Groups

    Zp×Zq\mathbb{Z}_p\times \mathbb{Z}_qZp​×Zq​ ≅\cong≅ Zpq\mathbb{Z}_{pq}Zpq​ if p and q are relatively prime

27. Characteristic of a Field