課程內容：



1. Group action. Burnside formula. Proof of Sylow theorems.

2. Free groups. Generators and relations. Characterization of groups of small orders

3. Basic concepts of fields. Algebraic and transcendental elements

4. Constructions with straight edge and compasses

5. Splitting fields. Algebraically closed fields. Zorn's lemma. Algebraic closure. Finite fields

6. Multiplicity of roots. Separable extensions. Purely inseparable extensions

7. Groups of field automorphisms. Galois groups

8. Normal extensions. Galois pairing

9. Symmetric rational expressions. Cyclotomic extensions

10. Solvable groups. Composition series. The butterfly lemma

11. Insolvability of the quintic



評分方式：

作業100%