Homework and Syllabus
Abstract Algebra 1
MATH 3140 Fall 2021
Homework must be submitted via Canvas, and is due at the start of class, with your name and homework number on it to receive credit. You will be graded on the clarity of your exposition. Messy, disorganized, or poorly written assignments will not receive credit. You must submit just one pdf file with all of the homework problems in the correct order.
You will receive 2 points Extra Credit on your homework for
doing it in LaTeX (you must upload both the .tex and
.pdf files to Canvas to get the extra credit).
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 the Math Academic Resource Center ("MARC" online) to be useful as a virtual meeting point for discussing homework, and a place to get some further help understanding the material.
An asterix * indicates that a homework assignment has not been finalized.
| Date | Topics |
Reading |
Homework |
| L1 Monday
August 23 |
Introduction to the
course, and overview of mathematical notation: A brief review of some mathematical notation, including sets, subsets, unions, intersections, products, maps ("functions"), and equivalence relations. This includes the following terminology for maps ("functions"): source ("domain"), target ("codomain"), image ("range"), injective ("one-to-one"), surjective ("onto"), bijective ("one-to-one and onto"). |
Fraleigh Section 0 You can also read a bit more about standard mathematical language in: Richard Hammack, The Book of Proof, Creative Commons, 2nd Edition, 2013. |
HW1a Fraleigh Exercises 0: 2, 4, 10, 12, 14, 17. |
| L2 Wednesday
August 25 |
Overview of mathematical
notation continued: Maps ("functions") and equivalence relations. |
Fraleigh Section 0 For those of you interested in learning LaTeX, please see the references at the bottom of this webpage. You may also want to take a look at The Not So Short Introduction to LaTeX 2e |
HW1b Fraleigh Exercises 0: 18, 19, 28, 30, 31, 36. |
| L3 Friday August 27 |
PART I: GROUPS AND
SUBGROUPS Introduction and examples: Introduction to how we think about abstract algebraic objects, and some examples we will use. |
Fraleigh PART I: Fraleigh Section 1 |
HW 1
DUE You must upload your homework to Canvas in .pdf format. Here is a LaTeX template for your homework, if you want it .pdf, .tex, .bib. Solution to 0.18 .pdf Solution to 0.30 .pdf HW2a Fraleigh Exercises 1: 4, 9, 12, 17, 37, 38. |
| L4 Monday August 30 |
Binary
operations and examples: Definition of binary operation, and examples. |
Fraleigh Section 2 | HW2b Fraleigh Exercises 2: 2, 6, 24, 26, 29, 36. |
| L5 Wednesday September 1 | Isomorphic
binary structures: Definition of isomorphism of binary structures, and examples. |
Fraleigh Section 3 |
HW2c Fraleigh Exercises 3: 4, 8, 12, 16, 26, 27. |
| L6 Friday September 3 | Review
homework |
HW 2
DUE Solution to 1.17 .pdf Solution to 2.2 .pdf Solution to 3.26 .pdf |
|
| Monday
September 6 |
LABOR DAY |
NO CLASS |
LABOR DAY |
| L7 Wednesday
September 8 |
Groups: Definition of group, and examples. |
Fraleigh Section 4 | HW3a Fraleigh Exercises 4: 2, 4, 11, 19, 29, 31. |
| L8 Friday September 10 |
Groups continued: Review of definition of a group, and more examples. |
Fraleigh Section 4, and Appendix:
Matrix Algebra You may also want to read the following brief review of linear algebra. |
HW 3 DUE Solution to 4.4 .pdf Solution to 4.31 .pdf HW 4a Fraleigh Exercises A (Appendix): 2, 6, 8, 10, 12, 14. |
| L9 Monday September 13 |
Subgroups: Definition of subgroup, cyclic subgroups, generators of cyclic subgroups, and examples. |
Fraleigh Section 5 | HW4b Fraleigh Exercises 5: 8, 14, 43, 45, 46, 54. |
| L10 Wednesday September 15 | Cyclic
groups: Elementary properties, structure, subgroups, and order of an element of a group. |
Fraleigh Section 6 Please read the following note on the order of an element of a group .pdf |
HW4c Fraleigh Exercises 6: 2, 5, 18, 30, 32, 50. |
| L11 Friday September 17 |
Generating
sets: Definition of a generating set and intersections of subgroups. |
Fraleigh Section 7 | HW 4 DUE Solution to A.14 .pdf Solution to 5.46 .pdf Solution to 6.50 .pdf HW5a Fraleigh Exercises 7: 2, 6. |
| L12 Monday September 20 |
Review |
||
| L13 Wednesday September 22 |
Review
practice exam |
Practice exam and
solutions |
|
| Friday September 24 |
MIDTERM
I |
MIDTERM I | |
| L14 Monday September 27 |
PART
II: PERMUTATIONS, COSETS, AND DIRECT PRODUCTS Groups of permutations: Permutation groups, symmetric groups, and Cayley's theorem. |
Fraleigh PART II: Fraleigh Section 8 |
HW5b Fraleigh Exercises 8: 9, 21, 30, 34, 35, 36, 44, 53. |
| L15 Wednesday September 29 |
Review
exam |
|
|
| L16 Friday October 1 |
Orbits,
cycles, and the alternating group: Orbits, cycles, length, parity, and alternating groups. |
Fraleigh Section 9 | HW 5
DUE Solution to 7.2 .pdf Solution to 8.30 .pdf HW6a Fraleigh Exercises 9: 2, 10, 29, 33, 34, 36. |
| L17
Monday October 4 |
Cosets and the theorem
of Lagrange: Left and right cosets, and the Theorem of Lagrange. |
Fraleigh Section 10 |
HW6b Fraleigh Exercises 10: 3, 12, 19, 35, 39, 40. |
| L18
Wednesday October 6 |
Finitely generated
abelian groups: Products of cyclic groups, order of elements, and the Fundamental Theorem of Finitely Generated Abelian Groups. |
Fraleigh Section 11 | HW6c Fraleigh Exercises 11: 10, 16, 24, 36, 46, 47. |
| L19 Friday October 8 |
PART III: HOMOMORPHISMS
AND QUOTIENT GROUPS ("FACTOR
GROUPS") Homomorphisms: Definition of a homomorphism, image, kernel and isomorphism. |
Fraleigh PART III: Fraleigh Section 13 |
HW 6 DUE Solution to 9.33 .pdf Solution to 10.3 .pdf Solution to 10.40 .pdf Solution to 11.47 .pdf HW7a Fraleigh Exercises 13: 2, 12, 13, 21, 47, 48. |
| L20 Monday October 11 |
Quotient
groups ("factor groups"): Normal subgroups, definition of the quotient group ("factor group"), and homomorphisms. |
Fraleigh Section 14 |
HW7b Fraleigh Exercises 14: 6, 10, 26, 30, 37, 39. |
| L21 Wednesday October 13 |
Quotient
group ("factor group")
computations, and simple groups: Simple groups, center, and commutator subgroup. |
Fraleigh Section 15 | HW7c Fraleigh Exercises 15: 8, 10, 15, 35, 36, 39. |
| L22 Friday October 15 |
Review
homework |
HW 7
DUE Solution to 13.2 .pdf Solution to 13.47 .pdf Solution to 14.30 .pdf Solution to 15.36 .pdf |
|
| L23
Monday October 18 |
PART IV: RINGS AND
FIELDS Rings and fields: Definitions, basic properties, homomorphisms, and isomorphisms. |
Fraleigh PART IV: Fraleigh Section 18 |
HW8a Fraleigh Exercises 18: 2, 9, 23, 32, 42, 46. |
| L24
Wednesday October 20 |
Integral domains: Zero divisors, cancellation, integral domains, characteristic. |
Fraleigh Section 19 | HW8b Fraleigh Exercises 19: 2, 6, 10, 15, 24, 29. |
| L25 Friday October 22 | Fermat's and Euler's
Theorems: Fermat's Theorem, Euler's Theorem, applications. |
Fraleigh Section 20 | HW 8 DUE Solution to 18.42 .pdf Solution to 19.10 .pdf HW9a Fraleigh Exercises 20: 2, 5, 8, 9, 23, 27. |
| L26 Monday October 25 | Review |
||
| L27 Wednesday October 27 |
Review
practice exam |
Practice exam
and solutions |
|
| Friday October 29 |
MIDTERM
II |
MIDTERM II |
|
| L28
Monday November 1 |
Fraction fields ("fields of quotients"): The fraction field ("field of quotients") of an integral domain. |
Fraleigh Section 21 | HW9b Fraleigh Exercises 21: 1, 2, 3, 4, 5, 6. |
| L29 Wednesday November 3 |
Review
exam |
||
| L30 Friday November 5 |
Polynomial rings: Polynomial rings in one variable, and evaluation homomorphisms. |
Fraleigh Section 22 | HW 9
DUE Solution to 20.27 .pdf Solution to 21.2 .pdf HW10a Fraleigh Exercises 22: 3, 7, 14, 24, 27, 28. |
| L31 Monday November 8 |
Factorization
of polynomials over a field: The division algorithm over a field, irreducible polynomials, uniqueness of factorization. |
Fraleigh Section 23 | HW10b Fraleigh Exercises 23: 2, 7, 14, 25, 34, 35. |
| L32 Wednesday November 10 |
PART
V: IDEALS AND QUOTIENT RINGS ("FACTOR
RINGS") Homomorphisms and quotient rings ("factor rings"): Homomorphisms, properties of homomorphisms. |
Fraleigh PART V: Fraleigh Section 26 |
HW10c Fraleigh Exercises 26: 1, 20, 21, 28, 29, 30. |
| L33 Friday November 12 | Homomorphisms
and quotient rings ("factor
rings") continued: Quotient rings ("factor rings"). |
Fraleigh Section 26 | HW 10
DUE Solution to 22.24 .pdf Solution to 23.34 .pdf Solution to 26.21 .pdf |
| L34
Monday November 15 |
Prime and maximal
ideals: Prime and maximal ideals, prime fields. |
Fraleigh Section 27 | HW11a Fraleigh Exercises 27: 2, 5, 31, 32, 34, 35. |
| L35
Wednesday November 17 |
Prime and maximal ideals
continued: Ideals in polynomials rings in one variable over a field, and unique factorization. |
Fraleigh Section 27 | HW11b Fraleigh Exercises 27: 14, 15, 16, 24, 26, 27. |
| L36 Friday November 19 |
Review
homework |
HW 11
DUE Solution to 27.2 .pdf Solution to 27.24 .pdf |
|
| November 22--26 |
THANKSGIVING
BREAK |
NO CLASS |
THANKSGIVING BREAK |
| L37
Monday November 29 |
PART VI: EXTENSION
FIELDS Introduction to extension fields: Introduction to extension fields, algebraic and transcendental elements, irreducible polynomials of elements, and simple extensions. |
Fraleigh PART VI: Fraleigh Section 29 |
HW12a Fraleigh Exercises 29: 3, 4, 7, 13, 23, 33. |
| L38 Wednesday December 1 | Vector spaces: Definition, linear independence and bases, dimension, and connection with field extensions. |
Fraleigh Section 30 | HW12b Fraleigh Exercises 30: 4, 11, 12, 15, 25, 26. |
| L39 Friday December 3 | Algebraic
extensions: Finite extensions and algebraic extensions. |
Fraleigh Section 31 | HW12 DUE Solution to 29.33 .pdf Solution to 30.4 .pdf HW13a Fraleigh Exercises 31: 1, 4, 14, 19, 24, 31. |
| L40 Monday December 6 | Review |
||
| L41 Wednesday December 8 | Review practice
exam |
Practice In-Class exam
and solutions Practice Take-Home exam and solutions |
HW13 DUE (note
the unusual date) Solution to 31.1 .pdf Solution to 31.33 .pdf |
| Friday
December 10 |
READING DAY |
NO CLASS |
READING DAY |
| Sunday December 12 | FINAL
EXAM 1:30
-- 4:00 PM |
FINAL EXAM |
You may want to use LaTeX
for typing homework. If you have a mac,
one possible easy way to get started is with texshop.
You will want to download the MacTex
package. 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. 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 sharelatex) for a cloud version.