# Rings and Fields

#### Final exam solutions

Here are solutions to the final exam.

I'll have packets of graded work ready for you to pick up by the beginning of winter term.

#### Office hours next week

My office hours next week will be:

• Monday, December 15, 10:00 a.m.–12:00 p.m.
• Tuesday, December 16, 10:00 a.m.–12:00 p.m.
• Wednesday, December 17, 2:00 p.m.–4:00 p.m.

I will also be checking my e-mail frequently throughout those three days. If you have questions about the take-home exam at a time other than office hours, that will be the best way to reach me.

IF YOU SUSPECT THAT THERE MIGHT BE AN ERROR ON THE FINAL EXAM, PLEASE LET ME KNOW AS SOON AS POSSIBLE.

#### A problem from this year's Putnam

Uusually Putnam problems have elegant solutions using material from the first two years of the college curriculum. I'm sure this one does too, but—as you'll suspect as soon as you read it—it's also highly vulnerable to a little Galois theory. (Here's a hint—but this would have been an entirely appropriate problem on Assignment 10, even without the hint.)

Problem B4
Let f(z) = az4 + bz3 + cz2 + dz + e = a (z- r1) (z- r2) (z- r3) (z- r4) where a, b, c, d, e, are integers, a ≠ 0. Show that if r1 + r2 is a rational number, and if r1 + r2r3 + r4, then r1r2 is a rational number.

#### What happened in lecture the Wednesday before Thanksgiving

We proved that A5 (the alternating group of even permutations of 5 elements) is simple, i.e., that it has no normal subgroups, by:

• proving that the group of rotational symmetries of a dodecahedron is simple, then
• proving that the group of rotational symmetries of a dodecahedron is isomorphic to A5, by considering the action on the 5 cubes hiding inside the dodecahedron.
See the third set of Artin handouts for details. (Unfortunately, I've already disassembled the dodecahedron model that got rotated and pointed at in class.)

#### Office hours before Thanksgiving

This week I'll hold office hours Monday, 2:30–4:30 and Tuesday, 3:30–4:30.

#### Solving cubic equations

If you really want to solve a cubic, this site will tell you how to do it. Several different ways.

#### Heron's formula

One of the solutions to a problem on Assignment 7 uses Heron's beautiful formula for the area of a triangle in terms of the lengths of its sides. Here's a link to several versions of the formula and a proof.

#### Presentation timeline

See the handout for more detail.

Wednesday, November 12
Deadline for telling me your topic and your partner(s). I will schedule talks as I hear what people are planning to do.
Two weeks before presentation (November 21, 24, or 26)
Preliminary outline of talk due. One page is fine; tell me what you think you want to cover and what sources you're using.
One week before presentation (December 1 or 3)
Detailed outline. This should probably be 3 pages or so, and should include precise statements of theorems, sketches of proofs, and summaries of contextual or historical material.
A day or two before your presentation (scheduled individually)
Rehearsal. I'll be the audience, and we'll probably be in the Math Library. This will be a full dress rehearsal; if you want to use Powerpoint or slides, or distribute a handout, you should have drafts prepared.
Friday, December 5, Monday, December 8 or Wednesday, December 10
Presentations, two per class hour.

You should come in and talk with me several times in the next month. At the very least, we should talk about your initial choice of topic and you should get feedback on your preliminary outline.

#### Another possible presentation topic

What's the real story on Galois? What mathematics did he do, and when did he do it, and how did he end up dying in a duel? E.T. Bell's chapter on Galois in Men of Mathematics is perhaps the most powerful presentation of the "myth" of Galois; there are plenty of other accounts that tell quite different stories.

#### Exam I Solutions, grading information

Here are solutions to Exam 1.

The mean score was 78, while the median was 88. An approximate curve might be:

• 90 or above: A range.
• 75 or above: B range.
• 50 or above: C range.

#### Exam I Rules

The first exam in Rings and Fields is about to happen. It will be a take-home exam covering Herstein, Chapter 3 and Section 5.1, along with other material discussed in lecture.

You may pick up the exam any time after 9:00 a.m. on Tuesday, October 14. You must return your exam no more than 24 hours after you pick it up, and by 4:30 p.m. on Friday, October 17.

Here are the rules that will be printed on the outside of the exam envelopes:

1. This exam is due back at the Mathematics Office, King 205, by 4:30 p.m. on Friday, October 17, 2003. You must return your exam during business hours: 8:30–noon or 1:00–4:30 p.m.
2. You must return your exam no more than 24 hours after picking it up.
3. There is a CONTROL SLIP inside this envelope. Fill this slip out—including your honor pledge—and staple it to your completed exam.
4. You must also sign the Class Log both when you pick up and when you return your exam.
5. You must work by yourself on this exam.
6. This exam is open book, open notes. You may consult Herstein and/or your own lecture notes as you work. You should not consult any other external resources, such as books, journals, web sites, or other human beings.

#### What's going on with old homework?

Short answer: I've been taking care of Math 220, who had their exam first.

Long answer: I've posted solutions to Assignment 6. There should be solutions to Assignment 5 by some point tomorrow (Monday), and I will try to have your 5's and 6's graded to pick up (from the Math Library) on Tuesday morning.

#### Imaginary Ideals Handout

On Monday, I gave out some pictures of ideals in various rings of algebraic integers contained in C.

#### Homework Solutions

I'll be posting the homework solutions as a single PDF document, one that will grow as new solutions are added. Each of the links on the left will take you to that omnibus document; I'll add new links as solutions are added.

I will also regularly update paper copies in the Math Library, King 203.

#### Departmental announcement

The Department of Mathematics employs a limited number of students to assist in correcting homework and tutoring students. The courses for which graders and tutors are sought range acrosss the whole mathematics curriculum but most particularly students are needed to help with calculus and statistics courses. These positions are a good way to solidify one's knowledge of the subject while also earning some money.

If you are interested, please fill out an application with the Mathematics Department secretary, Cathy Murillo, in King 205.

If you're in Rings and Fields, you should probably be at least considering grading or tutoring!

Also: the department only runs the evening walk-in tutoring program for calculus and statistics. If you're interested in individual tutoring, you should let Kay Knight in Peters 114 know. (Of course, those jobs don't become available until students request tutoring, typically a few weeks into the semester.)

#### Welcome to Rings and Fields!

We will meet on Mondays, Wednesdays, and Fridays from 1:30 p.m. to 2:20 p.m. in King 121.

Our textbook will be Herstein's Topics in Algebra. See the syllabus for more information.

Last updated December 18, 2003 by Elizabeth Wilmer.