# MATH 150C (Modern Algebra) – Spring 2021, UC Davis

Lectures: | $MWF$ 10:00-10:50am, 205 Olson (Anne Schilling) |

Discussion Sessions: | MAT 150C-001, CRN 29230, T 7:10-8:00 PM in CRUESS 107 |

Instructor: | Anne Schilling, MSB 3222, phone: 554-2326, [email protected] Office hours: Wednesdays 1-3pm |

T.A.: | Nate Gallup, MSB 2127 [email protected] Office hours: Thursday 4-6pm |

Text: | I will mostly follow Michael Artin, Algebra, published by Pearson, second edition, 2011. Another good reference is Dummit and Foote, Abstract Algebra, ISBN 0-471-36857-1. |

Pre-requisite: | MAT 150B |

Problem Sets: | There will be weekly $homework$ assignments due on Fridays at the beginning of class.You are encouraged to discuss the homework problems with other students. However, the homeworks that you hand in should reflect your own understanding of the material. You are NOT allowed to copy solutions from other students or other sources. If you need help with the problems, come to the discussion session and office hours! The best way to learn mathematics is by working with it yourself. No late homeworks will be accepted. A random selection of homework problems will be graded. |

Computing: | During class, I might illustrate some results using the open source computer algebra system Sage. When you follow the link, you can try it out yourself using Sage Online Notebook. Or you can sign up for a Class Account with the math department. Log into fuzzy.math.ucdavis.edu and type the command `sage` to launch a Sage session in the terminal. |

Exams: | There will be one Midterm on Wednesday May 7 in class. The Final exam will be Wednesday, June 11, 8-10am.There will not be any make-up exams! |

Grading: | The final grade will be based on: Problem sets 30%, Midterm 30%, Final 40%. Grades will be recorded on SmartSite. |

Web: | http://www.math.ucdavis.edu/~anne/SQ2021/mat150C.html |

Bed Time Reading: | If you would like some bedtime reading related to math, I can recommend two really good books by Simon Singh: “Fermat’s Last Theorem” and “The Code Book”. |

## Course description

This course is the third part of a three-quarter introduction to Algebra. Algebra concerns the study of abstract structures such as groups, fields, and rings, that appear in many disguises in mathematics, physics, computer science, cryptography, … Many symmetries ||can be described by groups (for example rotation groups, translations, permutation groups) and it was the achievement of Galois to distill the most important axioms (=properties) of groups that turn out to be applicable in many different settings. $We will discuss rings and fields, in particular the important concept of factorization in rings, and at the end discuss Galois$ theory.

The class is primarily based on Chapters 12, 14-16 of Artin’s book.

**1. Factorization**

factorization of integers and polynomials; unique factorization domains; principal ideal domains and Euclidean domains; Gaussian integers; primes; ideal factorization

**2. Modules**

definition of modules; matrices, ||free modules and bases; diagonalization of integer matrices; generators and relations for modules; structure theorem for Abelian groups; application to linear operators

**3. Fields**

examples; algebraic and $transcendental elements; field extensions$; finite fields; function fields; algebraically closed fields

**4. Galois Theory**

fundamental theorem of Galois theory; cubic equations; primitive elements; cyclotomic extensions

Best proof so far in class (thanks to Gwen!):