Homework and Syllabus
Abstract Algebra 1
MATH 3140 Summer 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 Tuesday
June 1 |
Introduction to the
course, and overview of mathematical notation: A brief review of some mathematical notation, including sets, subsets, unions, intersections, products, maps, and equivalence relations. |
Fraleigh Section 0 |
HW1a Fraleigh Exercises 0: 2, 4, 10, 12, 14, 17, 18, 19, 28, 30, 31, 36. |
| L2 Wednesday
June 2 |
PART I: GROUPS AND
SUBGROUPS Introduction and examples: Introduction to how we think about abstract algebraic objects, and some examples we will use. Binary operations and examples: Definition of binary operations, and examples. |
Fraleigh PART I: Fraleigh Section 1 Fraleigh Section 2 |
HW1b Fraleigh Exercises 1: 4, 9, 12, 17, 37, 38. HW1c Fraleigh Exercises 2: 2, 6, 24, 26, 29, 36. |
| L3 Thursday June 3 |
Isomorphic
binary structures: Definition of isomorphism of binary structures, and examples. Groups: Definition of group, and examples. |
Fraleigh Section 3 Fraleigh Section 4 |
HW1d Fraleigh Exercises 3: 4, 8, 12, 16, 26, 27. HW1e Fraleigh Exercises 4: 2, 4, 11, 19, 29, 31. |
| L4 Friday June 4 |
Groups
continued: Review of definition of a group, and more examples. Review of homework |
Fraleigh Section 4,
and Appendix: Matrix Algebra |
HW1 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 1.17 .pdf Solution to 2.2 .pdf Solution to 3.26 .pdf Solution to 4.29 .pdf Solution to 4.31 .pdf HW2a Fraleigh Exercises A (Appendix): 2, (6 optional), 8, 10, 12, 14. |
| L5 Monday June 7 |
Subgroups: Definition of subgroup, cyclic subgroups, generators of cyclic subgroups, and examples. Cyclic groups: Elementary properties, structure, subgroups |
Fraleigh Section 5 Fraleigh Section 6 Exercise (not for credit) on the order of an element of a group .pdf. |
HW2b Fraleigh Exercises 5: 8, 14, (43 optional), 45, 46, 54. HW2c Fraleigh Exercises 6: 2, 5, 18, (30 optional), 32, 50. |
| L6 Tuesday June 8 |
Generating
sets: Definition of a generating set, intersections of subgroups. PART II: PERMUTATIONS, COSETS, AND DIRECT PRODUCTS Groups of permutations: Permutation groups, symmetric groups, Cayley's theorem. |
Fraleigh Section 7 Fraleigh PART II: Fraleigh Section 8 |
HW2d Fraleigh Exercises 7: 2, 6. HW2e Fraleigh Exercises 8: 9, 21, 30, (34 optional), (35 optional), 36, 44, 53. |
| L7 Wednesday 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 |
HW2f Fraleigh Exercises 9: 2, 10, 29, 33, 34, (36 optional). HW2g Fraleigh Exercises 10: 3, 12, (19 optional), 35, 39, 40. |
| L8 Thursday June 10 |
Review | ||
| L9 Friday June 11 |
Review
of practice exam Review of homework |
Practice exam and solutions. | HW2 DUE Solution to 5.46 .pdf Solution to 6.50 .pdf Solution to 7.2 .pdf Solution to 8.30 .pdf Solution to 9.33 .pdf Solution to 10.3 .pdf Solution to 10.40 .pdf |
| Monday June 14 |
MIDTERM | MIDTERM | |
| L10 Tuesday June 15 |
Finitely
generated abelian groups: Products of cyclic groups, order of elements, the Fundamental Theorem of Finitely Generated Abelian Groups. PART III: HOMOMORPHISMS AND FACTOR GROUPS Homomorphisms: Definition of a homomorphism, image, kernel and isomorphism. |
Fraleigh Section 11 Fraleigh PART III: Fraleigh Section 13 |
HW3a Fraleigh Exercises 11: 10, 16, 24, 36, (46 optional), 47. HW3b Fraleigh Exercises 13: 2, 12, (13 optional), 21, 47, 48. |
| L11 Wednesday June 16 |
Review
of exam |
||
| L12 Thursday June 17 |
Factor
groups: Normal subgroups, definition of the factor group, quotient groups and homomorphisms. Factor group computations, and simple groups: Simple groups, center, and commutator subgroup. |
Fraleigh Section 14 Fraleigh Section 15 |
HW3c Fraleigh Exercises 14: 6, 10, 26, 30, (37 optional), 39. HW3d Fraleigh Exercises 15: 8, 10, 15, (35 optional), 36, 39. |
| L13 Friday June 18 |
PART
IV: RINGS AND FIELDS Rings and fields: Definitions, basic properties, homomorphisms, and isomorphisms. Integral domains: Zero divisors, cancellation, integral domains, characteristic. |
Fraleigh PART IV: Fraleigh Section 18 Fraleigh Section 19 |
HW3 DUE Solution to 11.47 .pdf Solution to 13.2 .pdf Solution to 13.47 .pdf Solution to 14.30 .pdf Solution to 15.36 .pdf HW4a Fraleigh Exercises 18: 2, 9, 23, 32, 42, (46 optional). HW4b Fraleigh Exercises 19: 2, 6, 10, 15, (24 optional), 29. |
| L14 Monday June 21 |
Fermat's
and Euler's Theorems: Fermat's Theorem, Euler's Theorem, applications. Fraction fields: The ring of fractions of an integral domain. |
Fraleigh Section 20 Fraleigh Section 21 |
HW4c Fraleigh Exercises 20: 2, 5, 8, (9 optional), 23, 27. HW4d Fraleigh Exercises 21: 1, 2, 3, 4, 5, (6 optional). |
| L15 Tuesday 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 Fraleigh Section 23 |
HW4e Fraleigh Exercises 22: 3, 7, 14, 24, 27, (28 optional). HW4f Fraleigh Exercises 23: 2, 7, 14, (25 optional), 34, 35. |
| L16 Wednesday June 23 |
PART
V: IDEALS AND FACTOR RINGS Homomorphisms and factor rings: Homomorphisms, properties of homomorphisms. Homomorphisms and factor rings continued: Factor rings. |
Fraleigh PART V: Fraleigh Section 26 |
HW4g Fraleigh Exercises 26: 1, 20, 21, 28, 29, (30 optional). |
| L17 Thursday 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 |
HW4h Fraleigh Exercises 27: 2, 5, 31, 32, 34, (35 optional). |
| L18 Friday June 25 |
PART
VI: EXTENSION FIELDS Introduction to extension fields: Introduction to extension fields, algebraic and transcendental elements, irreducible polynomials of elements, and simple extensions. Review of homework |
Fraleigh PART VI: Fraleigh Section 29 |
HW4 DUE
(deadline extended to Saturday June 26, 11:59 PM) Solution to 18.42 .pdf Solution to 19.10 .pdf Solution to 20.27 .pdf Solution to 21.2 .pdf Solution to 22.24 .pdf Solution to 23.34 .pdf Solution to 26.21 .pdf Solution to 27.2 .pdf HW5a Fraleigh Exercises 29: 3, 4, (7 optional), 13, 23, 33. |
| L19 Monday June 28 |
Vector
spaces: Definition, linear independence and bases, dimension, and connection with field extensions. Algebraic extensions: Finite extensions and algebraic extensions. |
Fraleigh Section 30 Fraleigh Section 31 |
HW5b Fraleigh Exercises 30: 4, 11, 12, 15, 25, (26 optional). HW5c Fraleigh Exercises 31: 1, 4, 14, (19 optional), 24, 31. |
| L20 Tuesday June 29 |
Review |
||
| L21
Wednesday June 30 |
Review of homework |
|
HW5 DUE (note
the unusual date) Solution to 29.33 .pdf Solution to 30.4 .pdf Solution to 31.31 .pdf |
| L22 Thursday July 1 | Review of practice
exam |
Practice In-Class
Exam and solutions. Practice Take-Home Exam and solutions. |
|
| Friday
July 2 |
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.sharelatex.com/
for a cloud version.