Monday 
Wednesday 
Friday 
Jan. 13
No class
 Jan. 15
In Lecture:
 Outline of the second semester
 Proof that if R is a UFD, so is R[x]; Gauss Lemma
 Heights of prime ideals, dimension of a commutative ring
Associated Reading:
 Lang, XII.6, II.5,
 DF, p. 232, 8.2, 8.3
 Hartshorne's "Algebraic Geometry," p. 5  7.
Video of class (downloadable)
 Jan. 17
In Lecture:

Jan. 20
No class, Martin Luther King holiday
 Jan. 22
Classes at Penn cancelled due to weather.
 Jan. 24 (9:00 a.m. in room 4C4 of DRL labs)
In Lecture:
 Dimension theory, continued
 Euclidean algorithm in a power series ring
Associated Reading:
 Hartshorne's "Algebraic Geometry," p. 5  7.
 Lang: 4.9
Video of class (downloadable)

Jan. 27
In Lecture:
 Weierstrauss preparation theorem
 Complete rings with respect to an ideal
 Start of the proof that k[[x_1,...,x_n]] is a UFD.
Associated Reading:
 Hartshorne's "Algebraic Geometry," p. 5  7.
 DF: 10.4  10.5
 Lang: 4.9, 3.3  3.4, 16.1  16.2.
Video of class (downloadable)
 Jan. 29
In Lecture:
 Completion of the proof that k[[x_1,...,x_n]] is a UFD.
 Bimodules
 Tensor products of modules
 Localization of modules
Associated Reading:
 DF: 16.2, 10.5
 Lang: p. 88, 3.4, 20.4, 26.3
 Hartshore's ``Algebraic Geometry," II.2, p. 143  145.
Video of class (downloadable)
 Jan. 31 (9:00 a.m. in room 4C4 of DRL labs)
In Lecture:
 Projective and injectives ( review) and flats
 Review of homological algebra, Ext and Tor
Associated Reading:
 DF: 16.2, 10.5
 Lang: p. 88, 3.4, 20.4, 26.3
 Hartshore's ``Algebraic Geometry," II.2, p. 143  145.
Video of class (downloadable)

Feb. 3
In Lecture:
 More on Tor
 Criteria for flatness
 The snake lemma
Associated Reading:
 DF: p. 788  795
 Lang: Chapter 16, sectiions 16.1  16.3
Video of class (downloadable)
 Feb. 5
In Lecture:
 Completion of the proof of the characterization of flatness via tensuring with ideals
 Grothendieck groups of modules,
 Start of Ktheory
Associated Reading:
 DF: 16.2, 10.5
 Lang: p. 88, 3.4, 20.4, 26.3
 Hartshore's ``Algebraic Geometry," II.2, p. 143  145.
 Milnor, "Introduction to algebraic Ktheory," chapter 1
Video of class (downloadable)
 Feb. 7 (9:00 a.m. in room 4C4 of DRL labs)
In Lecture:
 Local rings
 Dedekind rings
 Continuation of Ktheory
Associated Reading:
 DF: 12.1, p. 795
 Lang: 10.1, p. 171
 Hartshore's ``Algebraic Geometry," II.2, p. 143  145.
 Milnor, "Introduction to algebraic Ktheory," chapter 1
Video of class (downloadable)

Feb. 10
In Lecture:
 Dedekind rings, continued
 Locally free modules
Associated Reading:
 Hartshorne's "Algebraic Geometry," p. 5  7.
 DF: 10.4  10.5
 Lang: 4.9, 3.3  3.4, 16.1  16.2.
Video of class (downloadable)
 Feb. 12
In Lecture:
 Completion of some proofs about projective modules over Dedekind rings
 Noetherian modules and rings
 Hilbert basis theorem
Associated Reading:
 DF: chapter 12.1,
 Lang: chapter 10
 Milnor, "Introduction to algebraic Ktheory," chapter 1
Video of class (downloadable)
 Feb. 14
No class

Feb. 17
No class
 Feb. 19
No class
 Feb. 21
No class

Feb. 24
In Lecture:
 R[[x]] is Noetherian if R is commutative and Noetherian
 Associated prime ideals
 Primary decomposition
Associated Reading:
Video of class (downloadable)
 Feb. 26
In Lecture:
 Associated primes and primary decomposition, continued
Associated Reading:
Video of class (downloadable)
 Feb. 28
No class

March 3
In Lecture:
 Elliptic curves and the group law of Pic
 Proof that primary decompositions exist
Associated Reading:
Video of class (downloadable)
 March 5
In Lecture:
 The machinery needed to show primes associated to a primary decomposition are uniquely determined
 Associated primes of a module
Associated Reading:
Video of class (downloadable)
 March 7
No class

March 17
In Lecture:
 Radicals of annihilators, the support of modules, and the associated primes of modules
Associated Reading:
Video of class (downloadable)
 March 19
In Lecture:
 The fundamental theorem on finitely generated modules over a PID
Associated Reading:
Video of class (downloadable)
 March 21
No class

Mar. 24
In Lecture:
 Rational and Jordan canonical forms
Associated Reading:
 DF: 12.1, 12.3
 Lang: 5.1  5.3.
Video of class (downloadable)
 Mar. 26
In Lecture:
Associated Reading:
 DF: 12.3, 11.5
 Lang: 5.2. 26.7  26.8, 29.1
Video of class (downloadable)
 March 28
No class

Mar. 31
In Lecture:
 Symmetric and alternating tensors
 Determinants
Associated Reading:
 DF: 12.3, 11.5
 Lang: 5.2. 26.7  26.8, 29.1
Video of class (downloadable)
 April 2
In Lecture:
 Wedderburn's Theorem and group representation theory
 Start of field theory
 Straightedges versus rulers
 Algebraic closures
 Extending embeddings
Associated Reading:
 DF: 13.1, 13.2, 13.3
 Lang: 5.1, 5.2
Video of class (downloadable)
 April 4
No class

April 7
In Lecture:
 Ruler and compass constructions, continued
 Bezout's theorem
 Splitting fields
Associated Reading:
 DF: 13.3, 13.4
 Lang: 5.2
 Hartshorne: p. 47
Video of class (downloadable)
 April 9
In Lecture:
 Regular ngons and Fermat primes
 Existence of splitting fields and algebraic closures
 Normal, separable and Galois extensions
Associated Reading:
 DF: 13.5, 13.6, 14.1
 Lang: 5.1, 5.2, 5.3, 5.4, 5.6, 6.1
Video of class (downloadable)
 April 11
No class

April 14
In Lecture:
 Statement of the main theorems of Galois theory
 Examples
Associated Reading:
 DF: Chapter 14
 Lang: Chapter 5
Video of class (downloadable)
 April 16
In Lecture:
 Proofs of some of the main Theorems of Galois theory
 Determining Galois groups
Associated Reading:
 DF: Chapter 14
 Lang: Chapter 5
Video of class (downloadable)
 April 18
No class

April 21
In Lecture:
 S_n acting on k(x_1,...,x_n) by permutations,
 Discriminants
Associated Reading:
 DF: Chapter 14
 Lang: Chapter 5
Video of class (downloadable)
 April 23
In Lecture:
 Determining Galois groups of polynomials
 Example: Quartic extensions
 Using reductions of polynomials
Associated Reading:
 DF: Chapter 14
 Lang: Chapter 5
Video of class (downloadable)
 April 25
No class

April 28
In Lecture:
 Henselian rings, with applications to finding Galois groups
 Kummer Theory, Artin Schreier Theory
Associated Reading:
 DF: Chapter 14
 Lang: Chapter 5
Video of class (downloadable)
 April 30
In Lecture:
Associated Reading:
 DF: Chapter 14
 Lang: Chapter 5
Video of class (downloadable)
 May 2
No class
