Course Title:  Modern Algebra II
Course Number: MATH 421
Course Credits: 3 credits

Instructor: Dr. Janet Vassilev
Office: SMLC 324
Office Hours:  MW 11-11:30 am and F 1-3 pm or by appointment.
Telephone: (505) 277-2214
email: jvassil@math.unm.edu
webpage: http://www.math.unm.edu/~jvassil

Class Time: MWF 9-9:50 am
Class Location: Dane Smith Hall 227
Semester: Spring 2019

Course Description:  In addition to studying field extensions which are both algebraic and transcendental, we will learn advanced group theory and factorization properties of domains in order to better understand Galois Theory.

Course Goals:  This course is one of the fundamental courses for pure math students.  Not only will this course help you solidify your proof skills, but you will also strenghten and broaden your understanding of algebraic structures.

Student Learning Outcomes
• Write clear and concise proofs pertaining to algebraic structures.
• Use irreducible polynomials to construct field extensions.
• Find the splitting field of a polynomial.
• Determine if a field  extension is separable.
• Use the Sylow theorems to determine simplicity of a group.
• Find a composition series of a group.
• Find the Galois Group of a normal extension.
• Use the Galois Correspondence to determine all intermediate subfields of a field extension.
• Know the divisibility properties in different types of domains.

Text A First Course in Abstract Algebra, 7th Edition  By John Fraleigh.

Course Requirements

1) Daily Quizzes (100 points): The first 5 minutes of every class (excluding exam days) will consist of an open notebook quiz on the concepts of the previous lecture. The quizzes will be worth 5 points each. I will drop your 4 lowest quizzes and average the remaining quizzes to obtain a score out of 100.

2) Homework (200 points):  Homework will be assigned weekly on Wednesdays and will be collected the following Thursday by 8 am under my office door.  Homework will not be graded unless it is written in order and labeled appropriately.   The definitions and theorems in the text and given in class are your tools for the homework proofs. If the theorem has a name, use it. Otherwise, I would prefer that you fully describe the theorem in words, than state by Theorem 3.  Each week around 4 or 5 of the assigned problems will be graded. The weekly assignments will each be worth 20 points. I will drop your lowest two homework scores and the remaining homework will be averaged to get a score out of 200.

3) Exams (400 points):  I will give two midterms (100 points) and a final (200 points). There are no make up exams. If a test is missed, notify me as soon as possible on the day of the exam. For the midterms only, if you have a legitimate and documented excuse, your grade will be recalculated without that test using the percentage that you receive on the final exam.  The Midterms are tentatively scheduled for Monday February 26 and Monday, April 15.  The Final is on Wednesday, May 8, from 7:30-9:30 am.

GradesGeneral guidelines for letter grades (subject to change due to the class "curve"; but they won't get any more strict): 90-100% - A; 80-89% - B; 70-79% - C; 60-69% - D; below 60% - F.  In assigning Final Grades for the course, I will compare your grade on all course work (including the Final) and your grade on the Final Exam.  You will receive the better of the two grades.

Tentative Schedule (for Dr. Vassilev's Modern Algebra II):

 Date Section Topic Homework 1/14 27 Ideals in F[x] 1/16 29 Introduction to Extension Fields 27: 6, 8, 14, 30 32 29: 2, 4, 8, 16, 18, 23, 24, 25, 33 Due 1/24 1/18 29 Simple Extensions 1/23 30 Vector spaces over arbitrary fields 29: 29, 30, 31 30: 4, 6, 15, 21, 24, 25 Due 1/31 1/25 31 Algebraic Extensions 1/28 31 Algebraic Closures 1/30 31 Existence of the algebraic closure of a field 31:  3, 8, 12, 19, 23, 24, 26, 27, 29, 30, 31, 34 32: 2 Due 2/7 2/1 32 Geometric Constructions 2/4 33 Finite Fields 2/6 34 Isomorphism Theorems for Groups 33: 4, 8, 9, 12, 14 34: 3, 5, 8, 9 Due 2/14 2/8 35 Subnormal and Normal Series 2/11 35 Jordan Holder 2/13 36 Sylow Theorems 35: 4, 6, 8, 14, 17, 18, 22, 25, 26 36: 2, 4, 10, 11, 13, 18 Due 2/21 2/15 36 Sylow Theorems continued 2/18 37 Applying the Sylow Theorems 2/20 38 Free Abelian Groups 36: 15, 19, 22 37: 3, 4, 5, 7 38: 3, 8, 11 Due 3/4 In class 2/22 Review 2/25 Midterm 1 2/27 38 Proof of the Fundamental Theorem of finitely generated abelian groups 3/1 39, 40 Free groups and group presentations 3/4 45 PIDs and UFDs 3/6 45 PIDs are UFDs 39: 3, 5, 10, 12 40: 4, 8, 13 45: 10, 21, 25, 26 Due 3/21 3/8 45 More on UFDs 3/18 Class cancelled 3/20 46 Euclidean Domains 45: 29, 33, 34 46: 2, 12, 13, 15, 16, 17 Due 3/28 3/22 47 Gaussian Integers and multiplicative norms 3/25 48 Automorphisms of Fields 3/27 49 Isomorphism Extension Theorem 47: 3, 6, 8, 14 48: 4, 8, 12, 18, 29, 35 Due 4/4 3/29 49 Isomorphism Extension Theorem 4/1 50 Splitting Fields 4/3 50/51 Splitting fields and multiplicity 49: 2, 6, 8, 10, 13 50: 6, 9, 14, 18, 19 Due 4/11 4/5 51 Separable Extensions 4/8 51 Perfect Fields 4/10 52 Totally Inseparable Extensions 51: 2, 5, 9, 11, 12, 13 52: 1, 2, 5, 6, 7 Due 4/19 4/12 Review 4/15 Midterm 2 4/17 53 Galois Theory 53: 4, 6, 12, 15, 16, 17, 18(a,b), 19, 23, 24 Due 4/25 4/19 53/54 Galois Theory continued 4/22 54 Illustrations of Galois Theory 4/24 55 Cyclotomic Extensions 54: 4, 5, 7, 9, 10, 11, 13 55: 7, 9, 13 4/26 56 Insolvability of the quintic 4/29 56 Insolvability of the quintic 5/1 Review To turn in on day of final (5/8) 55: 14 & 56: 5. These are worth 40 points of the exam grade. 5/3 Review 5/8 Final Exam 7:30 am

Accomodation Statement: Accesibility Resource Center (Mesa Vista Hall 2021, 277-3506) provides academic support to students who have disabilities.  If you think you need alternative accessible formats for undertaking and completing your coursework, you should contact this service right away to assure your needs are met in a timely manner.

Title IX Statement: A note about sexual violence and sexual misconduct:  As a UNM faculty member, I am required to inform the Title IX Coordinator at the Office of Equal Opportunity (oeo.unm.edu) of any report I receive of gener discrimination which includes sexual harrassment, sexual misconduct, and/or sexual violence.  You can read the full campus policy regarding sexual misconduct.  If you have experienced sexual violence or sexual misconduct, please ask a faculty or starr memeber for help or contact the LoboRESPECT Advocacy Center.

Academic Integrity Statement: Collaborating with your peers can be an effective way to learn mathematics.  If you so choose to work with your peers, make sure that you each write up homework solutions in your own words.  Working with your peers and copying solutions is quite a separate matter.  If you attempt to copy someone's solution on the homework, you are only hurting yourself when it comes to taking the exams.  Struggling through a problem allows for much more understanding than just rewording someone else's argument.  You are more likely to solve the problems on in class exams, which are not a collaborative effort, if you have made an honest effort on the homework.  If a student attempts to cheat during an in class exam and I can prove this dishonesty, I reserve the right to take disciplinary actions against such a student which may result in a failing grade for that exam.