The lecture had a form of a review I made an effort to emphasize the main topics, ideas and concepts.
There will be no HW due for Monday, all you have to do is study for the test. The structure of the test will be similar to the first one. I.e. you have to answer x questions out of y (when y>x).
I feel it is very difficult to reproduce my lecture
in one page. Some of the main points are:
(A) Characterize when E/F is Galois in terms of F is the fixed field of the group Aut(E/F) and (for Char zero) E is a splitting field of a polynomial over F. The key for these are Theorems 7 and 9 of section 14.2.
Applications:
A1. The primitive element theorem (Theorem 25 of section 14.4)
just for characteristics zero
A2. Two proofs of the fundamental theorem of algebra.
(B) Splitting fields (existence and uniqueness) Theorems 25 and 27 from section 13.4.
For this we need Theorems 4,6, 8 of section 13.1
(C) Algebraically closed fields and algebraic closure: Existence and uniquness of algerbaic closure Propositions 30 and 31 of section 13.4 (notice that the book does not prove uniquness but I did in class!).
(D) Some easy facts:
D1. Proposition 33, 34 (of section 13.5) and thier use to prove
existence and uniqueness of finite fields.
D2. If F is a finite field of characteristics p then its Galois group
over the prime field is cylic (generated by the Frobenius
automorphism) of cardinality [F:F_p].
(E) Group theory:
E1. Sylow's first theorem
E2. if G is a p group acting on a finite set S then
|S| \cong |S_0| mod p where S_0 are the elements of S of orbit 1
E3. Conjugacy class eq theorem
E4. A finite p-group satifies the converse of Lagrange's theorem.
E5. Definition of solvable groups and the equivalent characterization
in terms of derived series (using commutator subgroups).
If G is solvable then also all its homorphic images are solvable.
E6. Definition of simple groups.
E7. p-groups are solvable.
I hope this will be of some value to you. Please note that the above list is incomplete. I made an effort to compress the more important aspects of the material between the first midterm to the last lecture. I definitely omitted many facts that are likely to be covered by the tyest. To find out what eactly happened in class you need to read your class notes.