Math 422/501 Lecture Plan

Main PageHomeworkExam info


Lecture schedule:
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.A-II.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 In-class 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 xn-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 Sn Jacobson 4.9, or Milne p. 75-79, Artin III.D
Mon, Oct 31 Building a rational polynomial with Galois group S5. Jacobson 4.10, or Milne p. 52-53
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.20-23 and 43-44, or Artin III.F.
Wed, Nov 16 More on constructibility
Fri, Nov 18 The primitive element theorem Jacobson p290-291, Milne p59-61, Artin II.M, Lang end of V.4.
Mon, Nov 21 The fundamental theorem of algebra; the discriminant of a polynomial Milne p61-62 (FTA); Jacobson 4.8, Milne p47-48
Wed, Nov 23 Computing the discriminant; Galois groups of cubics and quartics. Jacobson 4.8-4.9; Milne p49-52
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 p10-13.
Wed, Nov 30 Reduction mod p and Galois groups over the rationals. Jacobson 4.16 or Milne p. 54-57
Fri, Dec 2 Last day - TBD