Homework and Syllabus
Abstract Algebra 1
MATH 3140 Summer 2020
Homework is due at the start of class
and must submitted via CANVAS with your name
and homework number on it to receive credit. 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 your matrix computations. The program Mathematica,
for instance, is available free to students via
the University of Colorado.
An asterix * indicates that a homework assignment has not been finalized.
| Date | Topics | Reading |
Homework |
| Monday June 1 |
Introduction to the
course, and overview of mathematical notation: A brief review of some mathematical notation, including sets, subsets, unions, intersections, products, and maps. Matrix addition and multiplication, and determinant and inverse formulas for small matrices. |
Fraleigh Section 0 The Book of Proof, Richard Hammack, Creative Commons, 3nd edition, 2018. (Available for free at the author's website.) Judson Chapter 1 and Chapter 2 |
|
| Tuesday June
2 |
Review of linear
algebra: Vector spaces, linear maps, etc. Introduction to groups: Part I Examples of matrix groups and permutation groups. |
Some references to review linear
algebra are: Linear Algebra and its applications, by David C. Lay, 4th edition, Chapters 1--5. Calculus, Volume II, by Tom M. Apostol, 2nd edition. Fraleigh Appendix: Matrix Algebra Here is a short .pdf I wrote on vector spaces and linear maps. |
|
| Wednesday June 3 |
Introduction
to groups: Part II Abstract defintion of a group, subgroups, products of groups. |
Fraleigh Part I,
Fraleigh Sections 1-5 Judson Chapter 3 |
|
| Thursday June 4 |
Homomorphism
of groups: Definition of a homomorphism, image, kernel and isomorphism. |
Fraleigh Part III,
Fraleigh Section 13 Judson Chapter 11 |
|
| Friday June 5 |
Matrix
representations of groups: Definitions and basic properties Cyclic groups: Elementary properties, structure, subgroups |
Judson Chapter
12 Fraleigh Section 6 Judson Chapter 4 |
HW 1 Fraleigh Exercises 0: 2, 4, 10, 12, 14, 18, 28, 30, 36. Fraleigh Exercises A (Appendix): 2-14 even. Fraleigh Exercises 1: 4, 9, 12, 17, 37. Fraleigh Exercises 2: 2-24 even, 26-37. Fraleigh Exercises 3: 4, 8, 12, 16, 24-28. Here is a LaTeX template for your homework, if you want .pdf, .tex, .bib. Here is a sample solution to an exercise (not to be graded .pdf, .tex). |
| Monday June 8 |
Generating
sets: Definition of a generating set, intersections of subgroups. Groups of permutations: Permutation groups, symmetric groups, Cayley's theorem. |
Fraleigh Section 7 Fraleigh Part II, Fraleigh Section 8 Judson Chapter 5 You can also read the following note I wrote in class on Cayley's Theorem, as a demonstration of LaTeX .pdf, .tex. |
|
| Tuesday June 9 |
Orbits,
cycles, and the alternating group: Orbits, cycles, length, parity, alternating groups Cosets and the theorem of Lagrange: Left and right cosets, the Theorem of Lagrange. |
Fraleigh Section 9 Fraleigh Section 10 Judson Chapter 6 |
|
| Wednesday June 10 |
Finitely
generated abelian groups: Products of cyclic groups, order of elements, the Fundamental Theorem of Finitely Generated Abelian Groups. Factor groups: Normal subgroups, definition of the factor group, quotient groups and homomorphisms. |
Fraleigh Section 11 Judson Chapter 13 Fraleigh Section 14 Judson Chapter 10 |
|
| Thursday June 11 |
Review | ||
| Friday June 12 |
Review | Here are the notes from class
with solutions to Fraleigh Excercises 5: 46,
Excercises 6: 50, Excercises 8: 30, Excercises 9: 18,
Excercises 10: 3, Excercises 10: 40. |
HW 2 Fraleigh Exercises 13: 1-5 odd, 12, 13. Fraleigh Exercises 4: 2, 4, 11, 19, 29, 30, 31. Fraleigh Exercises 5: 1-3, 8, 13, 14, 43, 45, 46, 51, 54. Fraleigh Exercises 6: 5, 18, 30, 32, 50. Fraleigh Exercises 7: 2, 6. Fraleigh Exercises 8: 1-5 odd, 9, 21, 30, 34, 35. Fraleigh Exercises 9: 2, 10, 18, 23, 27. Fraleigh Exercises 10: 3, 5, 19, 35, 39, 40. Not for credit: Fraleigh Exercises 13: 21, 47, 48. Fraleigh Exercises 8: 36, 44, 49. Fraleigh Exercises 9: 29, 31, 33, 34, 36 (sample solution to 9.33 .pdf, .tex). Fraleigh Exercises 10:30, 31, 32, 33, 45. |
| Monday June 15 |
MIDTERM | MIDTERM | |
| Tuesday June 16 |
Review
of exam |
||
| Wednesday June 17 |
Factor
groups continued: Simple groups, center, and commutator subgroup. Rings and fields: Definitions, basic properties, and rings of matrices. |
Fraleigh Section 15 Fraleigh Part IV, Fraleigh Section 18 Judson Chapter 16.1 |
|
| Thursday June 18 |
Rings
and fields continued: Homomorphisms, isomorphisms, and fields. Integral domains: Zero divisors, cancellation, integral domains, characteristic. |
Fraleigh Part V, Fraleigh Section 26 Fraleigh Section 19 Judson Chapter 16.2 |
|
| Friday June 19 |
Fermat's
and Euler's Theorems: Fermat's Theorem, Euler's Theorem, applications. Fraction fields: The ring of fractions of an integral domain, and the total ring of fractions of a commutative ring |
Fraleigh Section 20 Judson Chapter 6.3 Fraleigh Section 21 |
HW 3 Fraleigh Exercises 11: 10, 16. Fraleigh Exercises 14: 5, 6. Fraleigh Exercises 15: 1, 4, 8,10. Fraleigh Exercises 18: 2, 9. Fraleigh Exercises 19: 1, 2, 6. Not for credit: Fraleigh Exercises 11: 17, 24, 36, 46, 47, 54 (sample solution to 11.47 and 13.47 .pdf, .tex). Fraleigh Exercises 14: 21, 26, 27, 30, 37, 38, 39 (sample solution to 14.30 .pdf, .tex) Fraleigh Exercises 15: 15, 34, 35, 36, 37, 39. Fraleigh Exercises 18: 23, 32, 33, 42, 46 (sample solution to 18.42 .pdf, .tex). Fraleigh Exercises 19: 10, 15, 24, 29. |
| Monday June 22 |
Polynomial
rings: Polynomial rings in one variable, and evaluation homomorphisms. Factorization of polynomials over a field: The division algorithm over a field, irreducible polynomials, uniqueness of factorization. |
Fraleigh Section 22 Judson Chapter 17.1 Fraleigh Section 23 Judson Chapter 17.2 |
|
| Tuesday June 23 |
Homomorphisms
and factor rings: Homomorphisms, properties of homomorphisms. Homomorphisms and factor rings continued: Factor and quotient rings. |
Fraleigh Part V,
Fraleigh Section 26 Judson Chapter 16.3 |
|
| Wednesday June 24 |
Prime
and maximal ideals: Prime and maximal ideals, prime fields. Prime and maximal ideals continued: Ideals in polynomials rings in one variable over a field, and unique factorization. |
Fraleigh Section 27 Judson Chapter 16.4 |
|
| Thursday June 25 |
Extension
fields: Introduction to extension fields, algebraic and transcendental elements, irreducible polynomials of elements, and simple extensions. |
Fraleigh Part VI,
Fraleigh Section 29 Judson Chapter 22.1 |
|
| Friday June 26 |
Algebraic
field extensions: Finite extensions and algebraic extensions. |
Fraleigh Part VI, Fraleigh Section 31 | HW 4 Fraleigh Exercises 20: 2, 5, 9, 23, 29. Fraleigh Exercises 21: 1, 4, 12, 13. Not for credit: Fraleigh Exercises 22: 3, 5, 7, 14, 23, 24, 27, 28. Fraleigh Exercises 23: 2, 7, 14, 16, 25, 34, 35. Fraleigh Exercises 26: 1, 4, 17. Fraleigh Exercises 26: 20, 21, 28, 29, 30, 34. Fraleigh Exercises 27: 2, 5, 24, 27, 31, 32, 33, 34, 35. |
| Monday June 29 |
Review |
||
| Tuesday June 30 |
Review |
||
| Wednesday
July 1 |
Review | ||
| Thursday
July 2 |
FINAL EXAM |
HW 5 Not for credit: Fraleigh Exercises 29: 3, 4, 7, 13, 23. |
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.sharelatex.com/
for a cloud version.