Syllabus

The course will progress tentatively as follows. We will begin by reviewing the main ideas of sets and functions, and from there we will begin our study of group theory. This should take roughly the first half of the course, after which we will have the midterm exam. We will then spend two to three weeks on ring theory. The course will end with some special topics and possibly applications of algebra, depending on the interests of the class. A more detailed schedule will be available on the course webpage, and any updates to the course structure will appear there.

Since the reading and writing of proofs is high on our list of objectives, many aspects of the course will be designed to develop these skills. For example, you will be strongly encouraged to read the textbook regularly. Also, the first few x-hours will be used as ``workshops'' for developing proof-writing skills. These will be purely for your benefit, and if you've already taken a proof-intensive class, you need not come regularly. After the first few weeks, we may continue the workshops if there is need, or we may begin using the x-hours to cover extra material and/or special topics.

The following is a tentative daily schedule for the course. It will be updated throughout the term to reflect our actual progress through the material. The reading assignment listed for each day is meant to be read after that day's class. It will generally be comprised of the sections in the book that we intend to cover in the next lecture, and it should be read before coming to the next class. There is no need to understand every detail of the assigned reading, but seeing it beforehand will help. All reading assignments are from Herstein unless otherwise noted.

Since I sometimes deviate from the textbook, or cover topics that are not fully addressed in the book, I will be posting a PDF version of my lecture notes here. The plan is to update them on a weekly basis.

Lecture Notes (Updated 8/17/2012)

We will be using the x-hour on Thursday, August 9th to cover polynomial rings.
There are review materials for the in-class midterm listed on the "Exams" page.
Here is the handout from the last proof workshop, on proofs by contraposition.
Here is the handout from the second proof workshop, which covers proofs by contradiction and mathematical induction.
Here is the handout from the first proof workshop, with some tips on constructing proofs and writing them up nicely.
The midterm has been moved to accommodate Dartmouth's family weekend. The in-class part of the exam will take place on Monday, July 23, and the take-home part will be due on Friday, July 27.
Here is a summary of the results surrounding the Euclidean algorithm from Saturday's class.
We will be meeting in the special Saturday class period on June 23, at 11:45 am.
Here is the set theory handout from the first day of class.
We will not meet in the first x-hour on June 21. We will use the x-hour on June 28 as a proof workshop.

Lectures	Brief Description	Reading Assignment
6/22	Overview: What is Abstract Algebra?	Set Theory handout and/or Section 1.2, skim Section 1.5
6/23	Motivating examples of groups	Section 2.1, Section 1.3
6/25	Definitions and examples of groups	Sections 1.4 and 2.1
6/27	More examples; the symmetric and dihedral groups	Section 2.2
6/28 (x-hour)	Proof Workshop
6/29	The symmetric and dihedral groups, Basic properties of groups	Section 2.3
7/2	Subgroups	No reading. (Optional: Section 4 of Gallian)
7/4	NO CLASS - Independence Day
7/5 (x-hour)	Proof Workshop
7/6	Cyclic Groups	Section 2.4 (up to the statement of Lagrange's theorem)
7/9	Cyclic groups (continued); Equivalence relations	Section 2.4 (beginning with Lagrange's theorem)
7/11	Cosets and Lagrange's Theorem	Section 2.5 (through the corollary to Theorem 2.5.5)
7/12 (x-hour)	Proof Workshop
7/13	Homomorphisms and isomorphisms	Section 2.5 (beginning with normal subgroups)
7/16	Cayley's Theorem and Kernels of Homomorphisms	Section 2.6
7/18	Kernels (cont.) and Quotient Groups	Section 2.7
7/19 (x-hour)
7/20	Quotient Groups (cont.) and Normal Subgroups	None
7/23	MIDTERM EXAM	Section 2.7
7/25	Quotient Groups and the First Homomorphism Theorem	Section 3.2 (Skim Section 3.1 if you want to review the basics of the symmetric group)
7/26 (x-hour)
7/27	The Symmetric Group: Cycle decomposition	Section 3.3
7/30	Even and Odd Permutations; The Alternating Group	Sections 2.9 and 2.10
8/1	Direct Products of Groups and The Fundamental Theorem of Finite Abelian Groups	Section 2.10
8/2 (x-hour)	The Fundamental Theorem of Finite Abelian Groups (cont.); the more general classification problem	Sections 4.1 and 4.2
8/3	Rings	Section 4.3
8/6	Ring Homomorphisms and Ideals	Section 4.4
8/8	Quotient Rings and Maximal Ideals	Section 4.5
8/9 (x-hour)	Polynomial Rings	Section 4.6
8/10	Irreducibility of Polynomials	Sections 5.3, 5.4 and 5.6
8/13	Roots of Polynomials and Field Extensions	Sections 5.3, 5.4 and 5.6
8/15	Field Extensions (continued)	None
8/16 (x-hour)	Presentations
8/17	Presentations; The Splitting Field of a Polynomial
8/20	Presentations
8/22	Recap and brief overview of Galois theory

Lectures

Brief Description

Reading Assignment

6/22

Overview: What is Abstract Algebra?

Set Theory handout and/or Section 1.2, skim Section 1.5

6/23

Motivating examples of groups

Section 2.1, Section 1.3

6/25

Definitions and examples of groups

Sections 1.4 and 2.1

6/27

More examples; the symmetric and dihedral groups

Section 2.2

6/28 (x-hour)

Proof Workshop

6/29

The symmetric and dihedral groups, Basic properties of groups

Section 2.3

7/2

Subgroups

No reading. (Optional: Section 4 of Gallian)

7/4

NO CLASS - Independence Day

7/5 (x-hour)

Proof Workshop

7/6

Cyclic Groups

Section 2.4 (up to the statement of Lagrange's theorem)

7/9

Cyclic groups (continued); Equivalence relations

Section 2.4 (beginning with Lagrange's theorem)

7/11

Cosets and Lagrange's Theorem

Section 2.5 (through the corollary to Theorem 2.5.5)

7/12 (x-hour)

Proof Workshop

7/13

Homomorphisms and isomorphisms

Section 2.5 (beginning with normal subgroups)

7/16

Cayley's Theorem and Kernels of Homomorphisms

Section 2.6

7/18

Kernels (cont.) and Quotient Groups

Section 2.7

7/19 (x-hour)

7/20

Quotient Groups (cont.) and Normal Subgroups

None

7/23

MIDTERM EXAM

Section 2.7

7/25

Quotient Groups and the First Homomorphism Theorem

Section 3.2 (Skim Section 3.1 if you want to review the basics of the symmetric group)

7/26 (x-hour)

7/27

The Symmetric Group: Cycle decomposition

Section 3.3

7/30

Even and Odd Permutations; The Alternating Group

Sections 2.9 and 2.10

8/1

Direct Products of Groups and The Fundamental Theorem of Finite Abelian Groups

Section 2.10

8/2 (x-hour)

The Fundamental Theorem of Finite Abelian Groups (cont.); the more general classification problem

Sections 4.1 and 4.2

8/3

Rings

Section 4.3

8/6

Ring Homomorphisms and Ideals

Section 4.4

8/8

Quotient Rings and Maximal Ideals

Section 4.5

8/9 (x-hour)

Polynomial Rings

Section 4.6

8/10

Irreducibility of Polynomials

Sections 5.3, 5.4 and 5.6

8/13

Roots of Polynomials and Field Extensions

Sections 5.3, 5.4 and 5.6

8/15

Field Extensions (continued)

None

8/16 (x-hour)

Presentations

8/17

Presentations; The Splitting Field of a Polynomial

8/20

Presentations

8/22

Recap and brief overview of Galois theory

Math 31: Topics in Algebra

Syllabus

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