Math 403 A: Introduction to Modern Algebra
Winter 2017

Instructor: Alexandru Chirvasitu

Class times and location:
Office hours: Mo 1 - 2:30, Tu 3:30 - 5, We 8 - 9 or by appointment
Email: chirva AT uw.edu

TA: Li Li
Office hours: Tu 10:30 - 12
Email: lil37 AT uw.edu

This is a continuation of the abstract algebra sequence beginning with Math 402, which focused on group theory.

In 403 we will be focusing on the general theory of a different type of mathematical structure: that of a ring. While groups were sets equipped with a binary operation meant to capture the abstract properties of composition for symmetries, a ring is a set equipped with two operations, mimicking addition and multiplication for any number of examples you have seen ``in nature'': the set of integers, or rational, or real, or complex numbers, the set of nxn matrices, the set of polynomials with real or complex coefficients, etc. etc.

There are several reasons why this is not a complete change of subject:
• First off, disregarding one of the two ring operations leaves you with what back in 402 we were calling an abelian group, so the general theory developed there for understanding such creatures now applies to the present setting.
• Secondly, in a broader sense, we are pursuing the same goals specific to abstract algebra as before: we are identifying some unifying structures and characteristics for certain mathematical objects of interest (like the ones I listed above; set of rationals, reals, matrices, etc.) and developing a general abstract framework for working with those structures. This then buys us the ability to apply that general theory simultaneously to the examples in question or any others one might discover or become interested in.

Textbook

We are (still) using Dan Saracino's Abstract Algebra (2nd edition). The textbook is absolutely necessary, both for the reading and in order to do the assignments.

It's essential that you do the reading. I won't have time to go over every relevant example in class, and you'll need a good grasp of the material in order to do the homework. In fact, I encourage you to regard the reading as part of your homework.

The phrase 'Section x', page numbers and result numbering (such as 'Theorem a.b') always refer to our textbook.

Due date Assignment Remarks
1 Fri Jan 06 Section 16 before
Theorem 16.1
We're not meeting that day.
2 Mon Jan 09 Finish Section 16
3 Wed Jan 11 Section 17
Stop on page 167 before
'In dealing with groups [...]'
4 Fri Jan 13 Continue Section 17
Stop on page 170
after Example 13.
5 Fri Jan 20 Finish Section 17
6 Wed Jan 25 Section 18
Stop before Theorem 18.3
Eeverything up to here
7 Mon Jan 30 study for the midterm; treat this
meeting as extended office hours / Q&A for the test
There will be no classes during the entire week of Feb 06.
8 Mon Feb 13 Continue Section 18
Stop before Theorem 18.8
9 Wed Feb 15 Finish Section 18
10 Fri Feb 17 Section 19
Stop before Theorem 19.2
11 Mon Feb 20 Continue Section 19
Stop after Example 3. on page 196
This is a university-observed
holiday; we're not meeting.
12 Wed Feb 22 Continue Section 19
Stop after Example 4. on page 198
13 Fri Feb 24 Finish Section 19
14 Mon Feb 27 Section 20
Stop before Theorem 20.3
15 Wed Mar 01 Finish Section 20
Use the rest of the quarter to study for the (cumulative) final.
16 Fri Mar 03 No meeting
17 Mon Mar 06
Wed Mar 08
In-class Q&A session for final
18 Fri Mar 10 Office hours in PDL C-417

Supplementary material

On occasion, I'll post extra notes, comments, etc. in this space.
• A description of the endomorphisms of the ring of rational numbers; this ties a loose end from the Wednesday, Jan 18 lecture.
• A solution for your last homework problem (20.12, page 210).

Homework

I am denoting the problems by x.y, as the textbook does (to mean problem y from Section x). Whenever you solve a problem, you can cite any preceding problem or theorem in your solution.

You have to turn in the assignment at the beginning of class, before the lecture. No late homework for any reason, but I will drop the lowest score.

Because of time constraints your TA will grade a couple of problems (tops) for correctness and the rest for completeness. I won't be telling you in advance which problems are graded for correctness though..

Due date Assignment Remarks
1 Wed Jan 11 16.1, 16.2, 16.3, 16.7, 16.12, 16.14
2 Wed Jan 18 16.16, 16.20, 16.23, 16.25
17.1, 17.2, 17.3, 17.5
3 Wed Jan 25 17.14, 17.19, 17.21, 17.27, 17.34, 17.35
18.1, 18.2, 18.3
Your homework problems up to here
are midterm prep.
No hw on Feb 01
because of the midterm
4 Wed Feb 15 18.5, 18.6, 18.14, 18.15
18.16, 18.18, 18.22, 18.25
5 Wed Feb 22 19.1, 19.4, 19.5, 19.6
19.10, 19.11, 19.16 (just parts (a) and (b) )
6 Wed Mar 01 19.2, 19.12, 19.13, 19.14
20.1, 20.2, 20.3, 20.4
7 Wed Mar 08 All remaining problems from Section 20

Midterm

It's happening in class (usual time and place):
The test is open book.

Final

The date and time are set by the department and I cannot change it. Early / late exams are not an option, so please plan accordingly.
Both the midterm and the final are open book exams. You get to bring your book, class notes, past homework, whatever (just do not collaborate with anyone during the tests).

• Homework : 20%
• Midterm : 35%
• Final : 45%

As mentioned above, we'll drop the lowest homework score.