Math 403B: Introduction to Modern Algebra

Winter 2018

Lectures: MWF 9:30-10:20 in CDH 125
Instructor: Jarod Alper (jarod@uw.edu)
Office: PDL C-544
Office hours: Wed 4-5, Fri 10:30-11:30

TA: Chi-yu Cheng ( chiyu@uw.edu )
Office hours: Tues 12:30-1:30, 3:30-4:30 in Padelford C-120/130
Textbooks and notes:
Syllabus: The course is the second part of a three-quarter sequence in modern algebra. We will cover simple groups, solvable groups, simplicity of A_n, introduction to rings, many examples of rings, ideals, principal ideal domains, and other selected topics.

Homework assignments: The lowest homework score will be dropped. Homeworks are due in the beginning of class.


Schedule: The content of the lectures will be posted here. The sections of the textbook quoted below are only a rough approximation of what we actually covered in class.

Lecture Date Topics covered Remarks
Week 1
1 Wed Jan 3 Introduction and simple groups (Goodman 10.1)
2 Fri Jan 5 Composition series and solvable groups (Goodman 10.1)
Week 2
3 Mon Jan 8 Solvable groups and properties of the alternating group (Goodman 10.1, 10.3)
4 Wed Jan 10 Simplicity of A_n for n >= 5 (Goodman 10.3)
5 Fri Jan 12 Introduction to rings (Judson 16.1) HW 1 due
Week 3
Mon Jan 15 No class--Martin Luther King Day
6 Wed Jan 17 Integral domains and fields (Judson 16.2)
7 Fri Jan 19 Ring homomorphisms and ideals (Judson 16.3)
Week 4
8 Mon Jan 22 Prime and maximal ideals, polynomials (Judson 16.4, 17.1) HW 2 due
9 Wed Jan 24 Polynomials: the division algorithm (Judson 17.2)
10 Fri Jan 26 Problem discussion
Week 5
11 Mon Jan 29 Midterm 1 HW 3 due
12 Wed Jan 31 Polynomials: k[x] is a PID (Judson 17.2-3)
13 Fri Feb 2 Polynomials: greatest common divisors and Eisenstein's criterion (Judson 17.2-3)
Week 6
14 Mon Feb 5 Problem discussion
15 Wed Feb 7 Gauss's Lemma and Eisenstein's criterion (Judson 17.3) HW 4 due
16 Fri Feb 9 Irreducible and prime elements in integral domains (Judson 18.2)
Week 7
17 Mon Feb 12 Discussion
18 Wed Feb 14 Euclidean domains, PIDs and UFDs (Judson 18.2) HW 5 due
19 Fri Feb 16 Field of fractions (Judson 18.1)
Week 8
Mon Feb 19 No class--Presidents' Day
20 Wed Feb 21 Discussion
21 Fri Feb 23 Midterm 2
Week 9
22 Mon Feb 26 More on UFDs (Judson 18.2)
23 Wed Feb 28 Proof that R UFD => R[x] UFD (Judson 18.2) and Applications of UFDs (Conrad) HW 6 due
24 Fri Mar 2 Solving y^2=x^3-2 (Conrad) and writing primes as the sum of two squares (Conrad, Theorem 9.6 )
Week 10
25 Mon Mar 5 Discussion
26 Wed Mar 7 Primes in the Gaussian Integers (Conrad)
27 Fri Mar 9 Pythagorean triples (Conrad) HW 7 due
Final
Wed Mar 14 Final examination, 8:30 - 10:20 am


Midterm: There will be two in class midterms
Final examination:
Grading: The lowest homework score will be dropped.