Monday | Wednesday | Friday |
---|---|---|

Jan. 9
- No class
| Jan. 11
In Lecture: - No class
| Jan. 13
In Lecture: - Outline of the second semester
- The fundamental theorem on finitely generated modules over a PID
Associated Reading: - D-F: 12.2
- Lang: 3.7, 3.5
Video of class (downloadable) |

Jan. 16
- No class, MLK holiday
| Jan. 18
We will have a class from 1:30 to 3:00 in room 3C2 of DRL labs - In Lecture
- Some open problems about P.I.D.s. Cohen Lenstra heuristics.
- The fundamental theorem on finitely generated modules over a PID.
Associated Reading:- D-F: 12.2
- Lang: 3.7, 3.5
Video of class (downloadable)
| Jan. 20
No class |

Jan. 23
In Lecture: - The basis theorem for vector spaces, Zorn's Lemma
- End of the proof of the fundamental theorem about finitely generated modules over a PID
Associated Reading: - D-F: 12.1, 12.3
- Lang: 5.1 - 5.3.
Video of class (downloadable) | Jan. 25
In Lecture: (room 3C2 of DRL labs) - Affine schemes
- Torsors
Associated Reading: - D-F: 12.3, 11.5
- Lang: 5.2. 26.7 - 26.8, 29.1
Video of class (downloadable) | Jan. 27
No class |

Jan. 30
In Lecture: - Rational canonical forms
Associated Reading: - D-F: 12.1, 12.3
- Lang: 5.1 - 5.3.
Video of class (downloadable) | Feb. 1
In Lecture: (room 3C2 of DRL labs) - More on canonical forms for endomorphisms of locally free modules over a ring
- Quiver algebras and the rational canonical form
Associated Reading: - D-F: 12.3, 11.5
- Lang: 5.2. 26.7 - 26.8, 29.1
- This Wikipedia article on quiver algebras
- This Note on moduli schemes
Video of class (downloadable) | Feb. 3
No class |

Feb. 6
In Lecture: - Representations of quiver algebras
- Fine and coarse moduli schemes
- Calculating rational canonical forms
Associated Reading: - D-F: 12.1, 12.3
- Lang: 5.1 - 5.3.
- This Wikipedia article on quiver algebras
- This Note on moduli schemes
Video of class (downloadable) | Feb. 8
In Lecture: (room 3C2 of DRL labs) - Calculating rational canonical forms (continued)
- Elementary divisors
- Jordan normal form
Associated Reading: - D-F: 12.3, 11.5
- Lang: 5.2. 26.7 - 26.8, 29.1
Video of class (downloadable) | Feb. 10
No class |

Feb. 13
In Lecture: - Start of field theory
Associated Reading: - D-F: 13.1, 13.2
- Lang: 5.1
Video of class (downloadable) | Feb. 15
In Lecture: - Distinguished extensions
- Straightedge and compass constructions
Associated Reading: - D-F: 13.3
- Lang: 5.1.
Video of class (downloadable) | Feb. 17
No class |

Feb. 20
In Lecture: - End of straightedge and compass constructions
- Algebraic closures
Associated Reading: - D-F: 13.3, 13.4
- Lang: 5.2.
Video of class (downloadable) | Feb. 22
In Lecture: - Extending embeddings
- Separable extensions
Associated Reading: - D-F: 13.3, 13.4, 13.5
- Lang: 5.3, 5.4
Video of class (downloadable) d | Feb. 24
No class |

Feb. 27
In Lecture: - Extensions of embedding of fields
- Hensel's Lemma
- Function fields
- Algebraic closures
Associated Reading: - D-F: 13.3, 13.4
- Lang: 5.2.
Video of class (downloadable) | Mar. 1
In Lecture: - Separability and normality of field extensions
Associated Reading: - D-F: 13.5, 13.6, 14.1
- Lang: 5.1, 5.2, 5.3, 5.4, 5.6, 6.1
Video of class (downloadable) | Mar. 3
No class |

March 6
No class - spring break! | March 8
No class - spring break! | March 10
No class - spring break! |

Mar. 13
In Lecture: - Separable extensions and polynomials
Associated Reading: - D-F: 13.5
- Lang: 5.4, 5.6.
Video of class (downloadable) | Mar. 15
In Lecture: - Separable extensions and polynomials, continued
- Inseparable extensions, inseparable degrees
Associated Reading: - D-F: 13.5
- Lang: 5.4, 5.6.
Video of class (downloadable) | Mar. 17
In Lecture: - Primitive element theorem
Associated Reading: - D-F: 14.4, 14.3
- Lang: 5.4. 5.5
Video of class (downloadable) |

Mar. 20
In Lecture: - End of proof of the primitive element theorem
- Finite fields
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) | Mar. 22
In Lecture: - Statement of the main theorems of Galois theory
- Examples
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) | Mar. 24
No class |

Mar. 27
In Lecture: - Galois groups as permutation groups
- Multiple transitivity of Galois groups
- The discriminant and the question of whether Galois groups lie in alternative groups
- Elementary symmetric functions and the canonical S_n action on F(x_1,...,x_n).
- Noether's problem.
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) | Mar. 29
In Lecture: - Explicit computations of some Galois groups
- Statement of the main theorem of Kummer theory
- Artin Schreier extensions
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) | Mar. 31
In Lecture: - Constructing S_n extensions of Q
- Abelian extensions of Q (statement of Kronecker Weber Theorem)
- Proof of the fundamental theorem of Galois theory for finite extensions
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) |

April 3
No class | April 5
In Lecture: - Proof of the fundamental theorem of Galois theory for infinite extensions
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) | April 7
No class |

April 10
In Lecture: - Cyclotomic extensions
- Irreducibility of cyclotomic polynomials over Q
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) | April 12
In Lecture: - Galois groups of composita of fields
- Representations of groups
- Linear independence of characters
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) | April 14
In Lecture: - Galois groups of compositums of fields
- Hilbert theorem 90 (Multiplicative and additive versions)
- Applications to Kummer theory and Artin Schreier theory
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) |

April 17
In Lecture: - Solvability by radicals
- Simple and semi-simple modules for rings
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) | April 19
In Lecture: - Semisimple modules and rings, continued
- Wedderburn's theorem
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
No Video of class - technical difficulties! | April 21
In Lecture: - Representations of groups
- Linear independence of characters
Associated Reading: - D-F: Chapter 14
- Lang: Chapter 5
Video of class (downloadable) |

April 24
In Lecture: Final exam, 1:30 - 3:30. | April 26
No class | April 28
No class |

Last updated: 4/18/17

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