Sebastian Casalaina

Homework and Syllabus

Abstract Algebra 1

MATH 3140 Fall 2022

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") to be useful as a 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 22
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 24
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 26
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 29
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 August 31 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 2 Review homework

HW 2 DUE

Solution to 1.17 .pdf

Solution to 2.2 .pdf

Solution to 3.26 .pdf




Monday September 5
LABOR DAY
NO CLASS
LABOR DAY
L7 Wednesday September 7
Groups:
Definition of group, and examples.
Fraleigh Section 4 HW3a

Fraleigh Exercises 4: 2, 4, 11, 19, 29, 31.
L8 Friday September 9
Groups continued:
Review of definition of a group, more examples, definition of isomorphism of groups.
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 12
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 14 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 16
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 19
Review


L13 Wednesday September 21
Review practice exam
Practice exam and solutions.


Friday September 23
MIDTERM I

MIDTERM I




L14 Monday September 26
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 28
Review exam


L16 Friday September 30
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 3
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 5
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 7
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 10
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 12
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 14
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 17
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 19
Integral domains:
Zero divisors, cancellation, integral domains, characteristic.
Fraleigh Section 19 HW8b

Fraleigh Exercises 19: 2, 6, 10, 15, 24, 29.
L25 Friday October 21 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 24 Review


L27 Wednesday October 26
Review practice exam
Practice exam and solutions.


Friday October 28
MIDTERM II

MIDTERM II




L28 Monday October 31
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 2
Review exam


L30 Friday November 4
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 7
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 9
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 11 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 14
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 16
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 18
Review homework

HW 11 DUE

Solution to 27.2 .pdf

Solution to 27.24 .pdf




 November 21--25
THANKSGIVING BREAK
NO CLASS
THANKSGIVING BREAK




L37 Monday November 28
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 November 30 Vector spaces:
Definition, linear independence and bases, dimension, and connection with field extensions.
Fraleigh Section 30

You may also want to read the following brief review of linear algebra again (from September 9)
HW12b

Fraleigh Exercises 30: 4, 11, 12, 15, 25, 26.
L39 Friday December 2 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 5 Review


L41 Wednesday December 7
Review practice exam
Practice exam and solutions.
HW13 DUE (note the unusual date)

Solution to 31.1 .pdf

Solution to 31.33 .pdf
Friday December 9
READING DAY
NO CLASS
READING DAY




Sunday December 11
FINAL EXAM 1:30 PM -- 4:00 PM
STAD 135 (Lecture Room)

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.