Date 
Topic 
Relevant reading 
Wed, Sept 7 
Course overview 

Fri, Sept 9 
Review of some ring theory 
Whatever textbook you used for ring theory. 
Mon, Sept 12 
First examples of field extensions 
N/A (none of the textbooks I've listed seem to have many examples
this early on; Chapter 13 of Dummit and Foote has a bunch if you
happen to have that book). 
Wed, Sept 14 
More review of ring theory + basic definitions for field theory 
Milne chapter 1 (through the section "Transcendental numbers") lines
up pretty well with what I'm going to talk about. (In other books
the material is roughly: Jacobson section 4.1, Artin sections
II.AII.C, Lang section V.1). 
Fri, Sept 16 
Some more examples of field extensions 
N/A (again, if you have Dummit+Foote, that has lots of
examples). 
Mon, Sept 19 
Algebraic extensions and splitting fields 
Jacobson 4.3, or Milne Ch. 2 (the ``splitting fields'' section), or
Artin II.D. (Lang V.3 could also work, but that's written more along
the lines of what I'll talk about on Friday). 
Wed, Sept 21 
More examples: Cyclotomic fields and finite fields 
All of the books have a section called "finite fields" and a section
called either "cyclotomic fields" or "roots of unity"; you can look at
any of those (though a lot are later on in the books so will get into
stuff we haven't talked about yet). 
Fri, Sept 23 
Algebraic closures 
Lang V.2 and Milne Ch. 6 both talk about algebraic closures  I
won't go into detail on proofs of existence and uniqueness,
though. 
Mon, Sept 26 
Normal extensions. 
Lang V.3 is closest to what I want to say about normal extensions
(the other books have it spread around). 
Wed, Sept 28 
Finish up on normality, and starting on separability 
For separable extensions: Jacobson 4.4, Milne's section called
"Multiple roots", or Lang V.4 
Fri, Sept 30 
More on separable extensions 

Mon, Oct 3 
Counting embeddings and automorphisms 
Basically Jacobson Theorem 4.4 (in section 4.3). This is the same as
Milne Proposition 2.7 (and I don't see in Artin or Lang in the context
I'm going to talk about). 
Wed, Oct 5 
Fixed fields of automorphisms 
The start of Jacobson Section 4.5, through the "Lemma on linear
equations". Equivalently, the first section of Milne Chapter 3, or the
beginning of Lang Section VI.1, or Artin sections II.F and the
beginning of II.H. (What we'll talk about here is the key idea Artin
used to give the modern formulation of Galois theory, actually). 
Fri, Oct 7 
Galois extensions 
Basically Jacobson Theorem 4.7 (in section 4.5). Equivalently the
second section of Milne Chapter 3. 
Wed, Oct 12 
The fundamental theorem of Galois theory 
The statement and proof of the theorem in Jacobson 4.7, Milne ch. 3,
Artin II.H, or Lang VI.1. 
Fri, Oct 14 
Some computations of Galois groups 
The examples sections in the books after the fundamental theorem,
and/or their sections on finite fields and cyclotomic fields. 
Mon, Oct 17 
Inclass Midterm Exam 

Wed, Oct 19 
Cyclotomic fields and permutation representations. 
Jacobson 4.8 and some of Jacobson 4.11 
Fri, Oct 21 
Composite fields 
The end of Lang VI.1. (Alternatively, if you have Dummit&Foote,
Section 14.4 of that is a good writeup). 
Mon, Oct 24 
Composite fields and direct products. 

Wed, Oct 26 
Composite fields and semidirect products, and the splitting field of
x^{n}d. 
Lang VI.9 (though we won't cover it as thoroughly as there). 
Fri, Oct 28 
The "general equation" of degree n has Galois group
S_{n} 
Jacobson 4.9, or Milne p. 7579, Artin III.D 
Mon, Oct 31 
Building a rational polynomial with Galois group S_{5}. 
Jacobson 4.10, or Milne p. 5253 
Wed, Nov 2 
Some group theory: Composition series and finite simple groups. 
Jacobson 4.6 (or appropriate sections of any group theory book). 
Fri, Nov 4 
More group theory: Solvable groups and simplicity of alternating
groups. 
Jacobson 4.6 (or appropriate sections of any group theory book). 
Mon, Nov 7 
Solvability by radicals 
Jacobson 4.7, Milne p.74, Artin III.C+III.E, or Lang VI.7. 
Wed, Nov 9 
Solvability by radicals continued 

Mon, Nov 14 
Constructible numbers and polygons 
Jacobson 4.2 and 4.11, Milne p.2023 and 4344, or Artin III.F. 
Wed, Nov 16 
More on constructibility 

Fri, Nov 18 
The primitive element theorem 
Jacobson p290291, Milne p5961, Artin II.M, Lang end of V.4. 
Mon, Nov 21 
The fundamental theorem of algebra; the discriminant of a
polynomial 
Milne p6162 (FTA); Jacobson 4.8, Milne p4748 
Wed, Nov 23 
Computing the discriminant; Galois groups of cubics and
quartics. 
Jacobson 4.84.9; Milne p4952 
Fri, Nov 25 
Galois groups and formulas for cubics and quartics. 
Jacobson 4.9 
Mon, Nov 28 
Finish up talking about quartics; reduction mod p and irreducibility
over the rationals. 
The irreducibility stuff should be review from your ring theory
course  see Jacobson 2.16 for instance, or Milne p1013. 
Wed, Nov 30 
Reduction mod p and Galois groups over the rationals. 
Jacobson 4.16 or Milne p. 5457 
Fri, Dec 2 
Last day  TBD 
