Homework and Syllabus
Abstract Algebra 2
MATH 4140/5140 Spring 2020
Homework is due in class and must be stapled
with your name and homework number on it to
receive credit. After Homework 3, your homework must
by typeset in LaTeX. You will be graded on the
clarity of your exposition.
Please read the suggested texts before class, and
then after class make sure to attempt the homework for the
sections we covered that day.
You may find it useful to use a computer algebra system to
check some of your computations. The program Mathematica,
for instance, is available free to students via
the University of Colorado.
You may find the Mathematics
Academic Resource Center ("MARC" MATH 175) to be
useful as a meeting point for discussing homework.
An asterix * indicates that a homework assignment has not been finalized.
| Date | Topics |
Reading |
Homework |
| Monday
January 13 |
Introduction |
Artin Chapter 1 |
|
| Wednesday
January 15 |
Introduction to LaTeX |
Read Chapters 1--2 of The Not So Short
Introduction to LaTeX 2e Bring a laptop with LaTeX installed (see the bottom of the page). We will use the following .tex and .bib files (which generated this .pdf) |
|
| Friday January 17 | Review: Groups |
Artin Chapter 2 |
HW 1 Artin Chapter 1 Exercises Section 1: 1.6, 1.9. Section 2: 2.8. Section 5: 5.2. Miscelaneous: M.3, M.4. |
| Monday January 20 |
MLK
DAY |
NO CLASS |
NO CLASS |
| Wednesday January 22 |
Review
continued: Vector spaces and groups of matrices. |
Artin Chapters 3 and
4 |
|
| Friday January 24 |
Rings:
Introduction, definition, polynomial rings. |
Artin Chapter 3
Section 3.2. Artin Chapter 11 Sections 11.1-2 |
HW 2 5 points extra credit for typing in LaTeX Artin Chapter 2 Exercises Section 5: 5.1, 5.2. Section 6: 6.3. Section 11: 11.8. Section 12: 12.2. |
| Monday
January 27 |
Rings continued:
Homomorphisms, ideals, and quotient rings. |
Artin Chapter 11 Sections 11.3-4 |
|
| Wednesday
January 29 |
Rings continued: Adjoining
elements, product rings. |
Artin Chapter 11 Sections 11.5-6 |
|
| Friday
January 31 |
Rings continued: Fractions. |
Artin Chapter 11 Section 11.7 |
HW 3 Artin Chapter 11 Exercises Section 1: 1.2, 1.8, 1.9. Section 2: 2.2. Section 3: 3.1, 3.5, 3.6, 3.7, 3.9, 3.12. Section 4: 4.1. |
| Monday February 3 |
Rings
continued: Maximal and prime ideals. |
Artin Chapter 11
Section 11.8 |
|
| Wednesday February 5 |
Rings
continued: Algebraic geometry |
Artin Chapter 11
Section 11.9 |
|
| Friday February 7 |
Review |
HW 4 Artin Chapter 11 Exercises Section 5: 5.1, 5.3, 5.5, 5.7. Section 6: 6.2, 6.3, 6.5. Section 7: 7.1, 7.2, 7.4. Not for credit (do not turn these in): Section 8: 8.2, 8.4. Section 9: 9.2, 9.3, 9.4. |
|
| Monday February 10 |
Review |
||
| Wednesday February 12 |
Review |
||
| Friday February 14 |
MIDTERM I | MIDTERM I |
MIDTERM I |
| Monday February 17 |
Review
Midterm I |
||
| Wednesday February 19 |
Factoring:
Introduction, factoring integers. |
Artin Chapter 12
Section 12.1 |
|
| Friday February 21 |
Factoring
continued: Unique factorization domains, Gauss' lemma. |
Artin Chapter 12
Sections 12.2-3 |
HW 5 Write solutions to the exam questions in LaTeX |
| Monday
February 24 |
Factoring continued:
Euclidean domains, principal ideal domains. |
||
| Wednesday
February 26 |
Factoring continued:
Factoring integer polynomials. |
Artin Chapter 12 Section 12.4 |
|
| Friday
February 28 |
Factoring continued: Gauss
primes. |
Artin Chapter 12 Section 12.5 |
HW 6 Artin Chapter 12 Exercises Section 1: 1.1, 1.3, 1.4(b). |
| Monday March 2 |
Quadratic
number fields: Introduction, algebraic integers. |
Artin Chapter 13
Section 13.1 |
|
| Wednesday March 4 |
Quadratic
number fields continued: Factoring algebraic integers,
ideals in $\mathbb Z[\sqrt{-5}]$. |
Artin Chapter 13
Sections 13.2-3. |
|
| Friday March 6 |
Quadratic
number fields continued: Ideal multiplication,
factoring ideals. |
Artin Chapter 13 Sections 13.4-5. | HW 7 Artin Chapter 12 Exercises Section 2: 2.2, 2.4, 2.9 Section 3: 3.2, 3.4, 3.5 |
| Monday
March 9 |
Quadratic number fields
continued: Prime ideals and prime integers, ideal
classes. |
Artin Chapter 13 Sections 13.6-7 |
|
| Wednesday
March 11 |
Quadratic number fields
continued: Computing the class group. |
Artin Chapter 13 Sections 13.8 |
|
| Friday March 13 | Quadratic
number fields continued: Real quadratic fields, and
lattices. |
Artin Chapter 13
Sections 13.9-10. |
HW 8 Artin Chapter 12 Exercises Section 4: 4.3, 4.6, 4.9 Section 5: 5.5, 5.7 |
| Monday March 16 |
Review |
||
| Wednesday March 18 |
Review |
||
| Friday March 20 |
Review
-- MIDTERM II cancelled |
MIDTERM II
cancelled |
|
| March 23--27 | SPRING BREAK |
SPRING BREAK |
SPRING BREAK |
| Monday
March 30 |
Review Midterm II |
||
| Wednesday
April 1 |
Fields:
Introduction, examples, algebraic and transcendental
elements. |
Artin Chapter 15 Sections 15.1-2 |
|
| Friday
April 3 |
Fields continued: The degree of a field extension, and finding the irreducible polynomial. | Artin Chapter 15 Sections 15.3-4 | HW 9 Artin Chapter 15 Exercises Section 1: 1.1, 1.2, 1.3 |
| Monday April 6 |
Fields continued: Ruler and compass constructions. | Artin Chapter 15 Section 15.5 | |
| Wednesday April 8 |
Fields continued: Adjoining roots, and finite fields. | Artin Chapter 15 Sections 15.6-7 | |
| Friday April 10 | Fields
continued: Primitive elements, function fields, and
the Fundamental Theorem of Algebra. |
Artin Chapter 15
Sections 15.8-10 |
HW 10 Artin Chapter 15 Exercises Section 2: 2.1, 2.3 Section 3: 3.1, 3.2, 3.3, 3.10 |
| Monday
April 13 |
Review |
||
| Wednesday
April 15 |
Galois theory: Introduction,
symmetric functions. |
Artin Chapter 16 Section 16.1 |
|
| Friday
April 17 |
Galois theory continued: The discriminant. | Artin Chapter 16 Section 16.2 | HW 11 Artin Chapter 15 Exercises Section 4: 4.1 Section 5: 5.1, 5.2, 5.4 Section 6: 6.1 Section 7: 7.1, 7.7 |
| Monday April 20 |
Galois theory continued: Splitting fields. | Artin Chapter 16
Section 16.3 |
|
| Wednesday April 22 |
Galois
theory continued: Isomorphisms of field
extensions and fixed fields. |
Artin Chapter 16
Sections 16.4-5 |
|
| Friday April 24 |
Galois
theory continued: Galois extensions and the main
theorem. |
Artin Chapter 16
Section 16.6-7 |
HW 12 Artin Chapter 15 Exercises Section 8: 8.1, 8.2 Not for credit (do not turn these in): Section 9: 9.1, 9.2, 9.3 |
| Monday
April 27 |
Review |
||
| Wednesday
April 29 |
Review |
||
| Friday May 1 | READING DAY |
NO CLASS |
NO CLASS |
| Tuesday
May 5 |
FINAL
EXAM 1:30 - 4:00 PM ECCR
116 (Lecture room) |
FINAL EXAM |
FINAL EXAM |
I strongly encourage
everyone to use LaTeX for typing homework. If you have
a mac,
one possible easy way to get started is with texshop.
If you are using linux,
there are a number of other possible ways to go, using
emacs, ghostview, etc. If you are using windows,
you're on your own, but I'm sure there's something online.
Here is a sample homework file to use: (the .tex
file, the .bib
file, and the .pdf
file). This site can help you find LaTeX symbols
by drawing: http://detexify.kirelabs.org/classify.html.
You may also want to try https://cocalc.com
(formerly https://cloud.sagemath.com/)
or https://www.overleaf.com
(formerly https://www.sharelatex.com/)
for a cloud version.