# MAT 535: Algebra II (Spring 2024)

### Announcements

- Lecture 2 was online; here are the notes.
- February 13 was a snow day.
- The first
**midterm**will be held in class on Thursday, March 7. - On Monday, March 4, there will be extra office hours from 10am to 4pm.
- Update: On
**Zoom**, you can find the link here. - Here are solutions to Midterm 1.
- The second
**midterm**will be held in class on Tuesday, April 9. - Here are solutions to Midterm 2.
- The
**final exam**will be held on Tuesday, May 7, from 8am to 10:45am in our usual classroom (Physics P–130). - Here are solutions to the final exam.

### About the course

Abstract algebra is the study of algebraic structures such as groups, rings, fields, or vector spaces. In the course, we will study several more advanced topics: Galois theory, basic multilinear and homological algebra, and an introduction to the representation theory of finite groups. You can find a more detailed list of topics on this page.

### Textbook

The textbook for the course is “Abstract Algebra” (3rd edition) by David S. Dummit and Richard M. Foote. Please see the official syllabus for additional information about the course, including university-wide policies.

### Grading

The final exam for the course will be on **May 7**. We are also
going to
have two midterms (in class) and weekly homework assignments. Your grade will
be determined based on the final exam (35%), the two midterms (20% each), and
your homework (25%). The general policy is no make-up exams and no late
homework, but there will be an extra homework assignment at the end of the
semester to make up for any missed assignments.

### Time and location

We meet on Tuesday and Thursday, 10:00–11:20 am, in room **Physics P130** (of the Physics Building).

### Office hours

My office hours are on Tuesday afternoon from 1:00pm–4:00pm.

### Schedule

Please read the corresponding sections before or after class.

Week | Chapters | Topics |

1 | 13.1, 13.2, 13.4 | Algebraic extensions, splitting fields |

2 | 13.5, 14.1 | Separable and normal extensions, automorphisms, Galois extensions |

3 | 13.6, 14.2, 14.3 | Finite Galois extensions, symmetric functions, roots of unity, finite fields |

4 | 14.2 | Galois correspondence |

5 | 14.4, 14.7 | Galois correspondence, composite extensions, solvability by radicals |

6 | 13.4, 14.8 | Galois groups over $\mathbb{Q}$, algebraic closures |

7 | 11.4, 12.2 | Characteristic and minimal polynomial, Cayley-Hamilton theorem |

8 | 12.3 | Jordan canonical form, diagonalization, normal operators |

9 | 11.3 | Bilinear forms, tensor product, wedge product |

10 | 10.3, 10.4 | Direct sum and tensor product of modules |

11 | 10.4 | Complexes, cohomology, exactness, free resolutions |

12 | 17.1 | Tor, long exact sequence, flat modules |

13 | 17.1 | Ext, projective modules, representations, Schur's lemma, complete reducibility |

14 | 17.3 | Characters, character table, representations of $S_n$ |

### Homework assignments

There will be a written homework assignment almost every week. Please write
up your solutions
nicely, staple all the pages together, and hand them in during the following
week, at the beginning of Thursday's class. Some of the problems will be graded ; the grader is **Xuande Liu**. We are also going to discuss some of
the problems in class on Thursday; you are expected to participate
in the discussion and, from time to time, volunteer to present a solution in
front of class.

Week | Assignment |

1 | Problem Set 1 (due Thursday, February 1) |

2 | Problem Set 2 (due Thursday, February 8) |

3 | Problem Set 3 (due Thursday, February 15) |

4 | Problem Set 4 (due Thursday, February 22) |

5 | Problem Set 5 (due Thursday, February 29) |

6 | Problem Set 6 (due Thursday, March 7) |

8 | Problem Set 7 (due Thursday, March 28) |

9 | Problem Set 8 (due Thursday, April 4) |

11 | Problem Set 9 (due Thursday, April 18) |

12 | Problem Set 10 (due Thursday, April 25) |

13 | Problem Set 11 (due Thursday, May 2) |

14 | Additional Problem Set (due Tuesday, May 7) |