Math 113 - Introduction to Abstract Algebra

InstructorPaul Vojta
LecturesTTh 12:30–2, Etcheverry 3107
Class Number22634
Office883 Evans
Office HoursTTh 10:30–12
Please email me to set up a time to meet if you cannot make any of these times.
PrerequisitesMath 54 or a course with equivalent linear algebra content
Required Text Fraleigh, A First Course in Abstract Algebra, 7th edition, ISBN: 9780201763904
Catalog DescriptionSets and relations. The integers, congruences, and the Fundamental Theorem of Arithmetic. Groups and their factor groups. Commutative rings, ideals, and quotient fields. The theory of polynomials: Euclidean algorithm and unique factorizations. The Fundamental Theorem of Algebra. Fields and field extensions.
Syllabus The course included the following sections of the textbook:
  • 0–11, 13–16, 18–23, 26, 27, 29–32, 45–46.
The course omitted:
  • Examples 7.10 and 7.12 (four properties satisfied by a Cayley digraph), and
  • The proof of Theorem 45.29 (which says that if D is a UFD, then so is D[x]).
GradingGrading will be based on:
20%Homework Assigned weeklyHomework is due in class, usually on Thursdays
20%First midterm Tuesday, October 1512:30–2:00 pm
25%Second midterm Tuesday, November 1912:30–2:00 pm
35%Final exam Friday, December 20 8:00–11:00 am
HomeworkAssigned weekly, generally due on Thursdays. Assignments are given below. Solutions will be posted on bCourses.
  • I tend to follow the book rather closely, but try to give interesting exercises and examples.
  • Note the final exam date given above (exam group 17). Do not enroll in this course if you cannot take the exam at that date and time, whether because of a conflict, too many exams on that day, or any other reason. The schedule of final exams is available at

Homework Assignments

General rules on homework assignments are:

Homework assignments are due in class on the days indicated below (handed in on paper). The lowest two homework scores are dropped.

No. Due Section Problems Comments
1 September 5 0 9, 12, 16, 17, 26, 32, 34
1 3, 8, 22, 25, 30, 32, 36
2 September 12 2 3, 6, 10, 22, 33, 36
3 2, 10, 16a, 27, 31, 33
3 September 19 4 2, 6, 24, 30, 36, 37
5 4, 11, 13, 16, 22, 23, 50
6 2, 6, 10, 18, 22, 34, 46
4 September 26 Download:    pdf    dvi
5 October 3 Download:    pdf    dvi
6 October 17 11 22, 39, 50, 51, 52
13 4, 8, 18, 44, 46, 50
14 31, 40
7 October 24 Download:    pdf    dvi
8 October 31 16 6, 9, 12, 13, 17
18 4, 6, 12, 18, 24, 38, 40, 42 In #12, you may assume that the square root of 2 is irrational.
19 10, 14, 23, 28, 29, 30 In #30, Z should be ℤ (referring to the ring of integers)
9 November 7 20 2, 6, 8, 12, 14, 27, 29
21 2, 6, 12, 13, 16
22 2, 6, 10, 14, 22, 25, 28
10 November 14 23 4, 6, 10, 16, 18, 20, 34, 37
26 4, 12, 14, 20, 21, 22, 28, 32
11 November 26 Download:    pdf    dvi
12 December 5 30 6, 10, 16, 20, 21, 24
31 8, 10, 23, 27, 29, 31, 33
45 2, 4, 10, 22, 25, 26, 30
13 Do not hand in
Solutions will be posted
on Thurs. Dec. 12
46 2, 4, 6, 10, 12, 17, 20, 21, 22
32 3, 5, 8, 9


We will use iClickers in class. These must be the physical iClickers, not the iClicker app. The latter is not allowed, because use of cell phones during class can be very distracting.

iClickers can be obtained from the bookstore, or you may be able to buy a used one from another student.

The grades from iClicker use will be incorporated into the homework portion of the course grade.

Clicker grades will be determined as follows. For each class day (other than the first two), you will receive one clicker point for participation if you have provided answers (correct or not) to at least 50% of the clicker questions on that day, and one clicker point for correctness if you answer at least one clicker question correctly on that day. These will be reported to bCourses separately, but will be combined for the purposes of the next steps.

I will then drop the lowest four sessions (class days), and then combine that information with the homework scores ... somehow. (This has yet to be determined, but it will be no more than 20% of the homework grade, assuming that such a percentage is meaningful. Sorry, I haven't done this before, so this is uncharted territory for me.)

The clicker points are not meant to be a big challenge. The vast majority of you should be receiving two clicker points on every day that you attend. The point of clickers is that you should be checking your own understanding of the material (including the reading) as the course progresses.

The clicker grading plan is subject to change.

Course Handouts

No.DateTitle Download
1August 29 Equivalence relations and partitions pdf dvi
2September 17 These are some of our favorite groups pdf dvi
3September 17 On xn pdf dvi
4October 3 Details on Proof II of Theorem 9.15 pdf dvi
5November 7 𝑛 is a ring and γ:ℤ→ℤ𝑛 is a ring homomorphism pdf dvi
6November 12 Gauss's Lemma and a corollary (revised 11/14) pdf dvi
7December 5 A slicker proof that all PIDs are UFDs pdf dvi

Lecture Slides

Slides from lectures will be posted here, generally on the same day after the lecture has ended. Not all lectures will have slides, and slides don't necessarily cover the whole lecture.

Here are the slides that have been posted so far:

Thu. Oct. 24 Original format Letter format Section 18: Rings and Fields
Tue. Oct. 29 Original format Letter format Finite rings, and a homomorphism γ:ℤ→R
Thu. Oct. 31 Original format Letter format Section 20: Fermat's and Euler's Theorems (and some congruences)
Tue. Nov. 5 Original format Letter format Sections 21–23: Fields of quotients, polynomials, and division algorithm for F[x]
Thu. Nov. 7 Original format Letter format Sections 23 and 26: Factorizing in F[x] and ring homomorphisms
Tue. Nov. 12 Original format Letter format Sections 26 and 27: Ideals, factor (quotient) rings, and prime and maximal ideals
Thu. Nov. 14 Original format Letter format Sections 27 and 29: Ideals in F[x], the ``Basic Goal,'' and extension fields
Thu. Nov. 21 Original format Letter format Section 30: Linear algebra (not the whole lecture)
Thu. Dec. 5 Original format Letter format Sections 45 and 46: Gcds in integral domains, and the Euclidean Algorithm (not the whole lecture)

Exams (Generally)

Policies for exams are as follows.

The two midterms will be given during the normal class hours (12:30–2 pm), and will be in our normal classroom (Etcheverry 3107).

Generally, about a week before each exam, a sample exam will be distributed in class and posted on bCourses. This will usually be an exam from an earlier Math 113 class that I've taught. Sample exams should be used to get a general idea of the likely length of an exam and the general nature of questions to be asked (e.g., the balance between computational and more theoretical questions). However, one should not (for example) observe that a sample exam contains questions on material from Sections 1.5, 2.1, 2.7, 3.1, 3.4, etc., and expect to see questions from those sections on the actual exam.

Exams are cumulative, so the second midterm may have questions from material prior to the first midterm. Of course, the final exam will cover the whole course, but will have increased emphasis on the material not covered on the midterms.

Here is a link "How to lose marks on math exams" (by a former GSI Andrew Critch).

The Math Department maintains an archive of old exams (usually without answers). Here is the link for Math 113.

And finally, a word about regrades: Grade calculation errors are welcome for discussion or review. Whether this solution should be worth 4 or 5 points is not.

First Midterm

Note: The scope of the midterm was reduced, due to the class cancellation of October 10.

The first midterm was given on Tuesday, October 15, from 12:40 to 2:00 pm, in our usual classroom. It covered:

It did not cover:

A sample midterm was distributed in class on October 8. Both the sample midterm and solutions to it are available on bCourses.

A dummy copy of the sample midterm in the format to be used on the actual exam is also available on bCourses (under the name “Sample First Midterm (Booklet Version)”). The instructions on the first page of that exam are the same as will be on the actual exam. Please familiarize yourselves with them. The instructions for the first two problems on the actual exam will also be identical to those on the sample midterm. (Of course, the definitions and examples sought will be completely different, as will the remaining questions on the exam.)

Finally, a list of definitions in the course so far is available on bCourses. Be sure to know them all—you don't want to be in a position where you don't even know what a question is asking! (This list has also been revised to reflect the reduced scope of the midterm.)

Solutions to the midterm itself are also now available on bCourses. The (very very rough) curve for the midterm is:


The median was 46, the mean was 47.8, and the standard deviation was 22.4.

The distribution of scores was as follows.

Histogram of Midterm Scores

Second Midterm

The second midterm was given on Tuesday, November 19, from 12:40 to 2:00 pm, in our usual classroom. It covered:

It did not cover:

A sample midterm was distributed in class on November 12. The sample midterm and solutions for it are available on bCourses.

Solutions to the midterm itself are also now available on bCourses. The (rough) curve for the midterm is:


The median was 86.5, the mean was 81.6, and the standard deviation was 29.6.

The distribution of scores was as follows.

Histogram of Midterm Scores

RRR Week

We will continue to meet on both Tuesday and Thursday of RRR week (at the usual time and place). We will be doing review.

Office hours will also continue on their usual schedule, for both RRR week and final exam week, except that the office hour for Tuesday, December 10 will only be 11:10–12:00. This is because of a meeting.

Final Exam

A sample final exam was distributed in class on Tuesday, 10 December. It is also available on bCourses, as are solutions for it.

The final exam was given on Friday, December 20, 8–11 am, in our usual classroom (Etcheverry 3107). It covered the whole course (all of the reading, homework problems, and lectures). The list of sections covered is listed in “Syllabus” above (it has been updated now that the lectures have all been given). See also the Review Sheet and lists of definitions posted on bCourses.

Approximately 45% of the exam was on new material not covered on either of the midterms.

The median was 114.5, the mean was 118.4, and the standard deviation was 35.6.

Last updated 27 December 2019