ALGEBRA 3

After passing this exam, the student should be able to 1. Define the basic notions and theorems about group actions and permutation groups. 2. Prove that S_n and A_n are simple groups for n>4. 3. Consider field extension as a vector space, and to conduct the algorithm for field extension. 4. To prove the Kronecker's theorem on the existence of the splitting field and to know its consequences. 5. Understand group of automorphisms of a field, Galois group and the correspondence between lattice of subgroups and lattice of subfields. 6. Prove that the fifth degree polynomials are not solvable by radicals. 7. Understand how Galois theory solves the classic problem of "solvability of equations by radicals" over some fields.