Modern Algebra for Hua Class

1. Introduction

2. Sets and Maps

3. Basics on Rings

4. Homomorphisms and Ideals

5. Two Constructions of Fields

6. Polynomials in One Variable

7. Euclidean Domains

8. The Ring of Guassian Intergers

9. UFD

10. Simple Field Extensions

11. Algebraic Field Extensions

12. Splitting Fields

13. Finite Fields

14. Cyclotomic Fields

15. Groups and Lagrange’s Theorem

16. Cyclic Groups

17. Normal Subgroups

18. Symmetric Groups

19. Group Actions

20. Sylow Subgroups

21. Presentations of Groups

22. Finitely Generated Abelian Groups

23. Galois Extensions

24. The Fundamental Theorem of Galois Theory

25. The Fundamental Theorem of Algebra

26. Radical Extensions and Solvable Groups

27. Galois’s Great Theorem

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Updated on May 27, 2025