# Teaching / Fundamental Concepts of Modern Algebra

**Disclaimer:** In-person lectures. No recordings, solutions, or make-up tests.

## Weekly Schedule

Asterisks indicate short weeks or weeks I am away. Exclamation marks indicate test weeks.

Week | Reading | Deliverable | Section |
---|---|---|---|

1 | Rings, integral domains, order (MB 1.1-1.4) | 1.2#1,3,5,7,8b; 1.3#1-4,7; 1.4#1,2 | 1.4#3-5, 1.5#6,8 |

2* | W.o.p., induction (MB 1.5-1.8) | 1.5#1,2,5(a,b, or c),11-12,14; 1.7#1,3,5,6,10,14 | 1.4#3-5, 1.5#6,8 |

3 | Euclidean algorithm, FTA, valuation, congruences (MB 1.9-1.12) | 1.8#1,5,8; 1.9#6,11,16 | |

4 | Maps of sets, fields (MB 1.11-1.12, 2.1-2.4) | 1.11#2, 1.12 #3, 4, 2.1 #2, 6, 9 | 1.10#7 |

5 | Detour on p-adic integers (Cuoco, Koblitz Ch 1) | 2.3#3, 2.4#2, 3.1#7; 2.5-2.6 in Perusall | |

6! | Polynomials (MB 3) | Test 1 on Ch 1 & 2 Feb 13 | |

7* | UFDs (MB 3) | 3.1#3, Prove the corollary of 3.2 theorem 4. 3. 3.2#1, 3.2#2, 3.3 # 1(c), 3.3#4 | 3.2#9 |

8 | Extension fields and their automorphism groups (MB 6, 15) | 3.3 #3,7; 3.5 #2,3,4,5 | 3.5#6 or 7 |

9 | Groups (MB 6) | 6.1#3, 6.2#3,4 | Dihedral groups |

March | Break March Break ... | March Break March ... | Break |

15 | Death by a thousand cuts | TODO | TODO |

11* | Actions and the orbit-stabilizer theorem | TODO | TODO |

12* | Normal subgroups and quotient groups | TODO | TODO |

13 | Symmetric and alternating groups, and the platonic solids | TODO | TODO |

14! | Center, centralizers, conjugation, simplicity of |
TODO | Test 2 on TBA Apr 17 |

15 | Automorphisms (inner, outer) | TODO | TODO |

## Specs

**Instructor**: Anne Dranowski (dranowsk@usc.edu)

**Office**: KAP 444C or my zoom room**Office hour windows**: Tuesdays 3:45-5 in Zoom and Fridays 2:30-4:30 in person (as much as possible, please email to confirm your attendance)

**TA**: Kayla Orlinsky (korlinsk@usc.edu)

### Class schedule

All ~five hours of weekly classes are equally important. As much as possible we will spend class time motivating difficult concepts or elaborating on harder examples or walking through a proof together (as opposed to, say, regurgitating the details of an easy proof that can be found in the book and that you can reconstruct on your own)

**Lectures**: Mondays, Wednesdays, and Fridays 1:00-1:50pm in CPA 211**Discussion**: Tuesdays 2:00-3:50pm in GFS 222**Test 1**: approx. February 13th, in class**Test 2**: approx. April 17th, in class**Project**: Sign up for 1-2 5-30 minute planning meeting(s) here**Final exam**: Wednesday, May 3rd, 2:00-4:00 p.m. Please review USC Policy here: https://classes.usc.edu/term-20231/finals/.

### Resources

**The Textbook**: A Survey of Modern Algebra by Garrett Birkhoff & Saunders Mac Lane (4th Ed.)**Lectures & Discussions**: Some of the material we may uncover (e.g. Sylow's theorems) will be outside the scope of the book. If it is significant, I will let you know**Math Center**: The math department hosts a math help center. See their website for details**Accessibility**: Please contact OSAS as soon as possible should you require accommodation**Calendar**: Session dates are summarized here https://classes.usc.edu/term-20223/calendar/

### Platforms

- Course material will be posted to the course webpage and communicated thru Blackboard announcements, therefore please check Blackboard announcements regularly.
- Homework assignments and other assessments will be administered in Gradescope, which can be accessed through Blackboard.

## Content objectives

This course will provide a detailed introduction to modern abstract algebra, which is a basic part of the language of much of modern math.

Time permitting we aim to cover parts of Chapters 1-3, 6, 9-11, and 13-15.

## Course structure

Beyond classroom time and the textbook, the course will involve:

**Reading:** Weekly readings will be assigned from the textbook, corresponding to the material that we are uncovering. You are expected to read the entirety of a chapter to understand it, in addition to class notes, even if we do not complete all the details in class.

**Homework:** There will be roughly weekly homework assignments. Late assignments will not be accepted: this course is unforgiving if you fall behind, so I do not want to encourage it. The three lowest scores will be dropped for every student.

**Project:** You will select one of the starred problems from the textbook and write up a detailed solution in the form of a short math article, preferably in LaTeX. Collaboration is welcome.

The final grade will consist of

- homework (13%)
- project (18%)
- two tests (18% each)
- cumulative final exam (33%)

**Additional References**: