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Contents

Crucial Information

Agenda: Follow Évariste Galois to the top of mathematics' first mountain.

Classes: Wednesdays 6-9PM (OMG) at Sidney Smith 1086.

Instructor: Dror Bar-Natan, drorbn@math.toronto.edu, Bahen 6178, 416-946-5438. Office hours: by appointment.

Teaching Assistant: Yichao Zhang, zhangyichao2002@hotmail.com. Office hours: Tuesdays 1-3 at the Math Aid Centre, Sidney Smith 1071.

Abstract

Taken from the Faculty of Arts and Science Calendar:

Commutative rings; quotient rings. Construction of the rationals. Polynomial algebra. Fields and Galois theory: Field extensions, adjunction of roots of a polynomial. Constructibility, trisection of angles, construction of regular polygons. Galois groups of polynomials, in particular cubics, quartics. Insolvability of quintics by radicals.
• Prerequisite: MAT224H1 (Linear Algebra II), MAT235Y1/MAT237Y1 (Calculus II/Multivariable Calculus), MAT246H1/MAT257Y1 (Concepts in Abstract Mathematics/Analysis II).
• Exclusion: MAT347Y1 (Groups, Rings and Fields).

Text Book(s)

• (Required) J. A. Gallian, "Contemporary Abstract Algebra", chapters 12-17, 20-23 and 31-33 (approx.).
• (Recommended) D. S. Dummit and R. M. Foote, "Abstract Algebra", chapters 7, 8, 9, 13, 14.
• (Suggested) T. Hungerford, "Abstract Algebra, an Introduction".

Plan

I will aim to cover the above-mentioned 13 chapters of Gallian's book at a bit faster than one per week, so as to leave us some time for extras at the end, but we may fall back to a rate of just one chapter a week or even less. If so, chapters 23 and 31 will be the first candidates for skipping.

Wiki

The class web site is a wiki, as in Wikipedia - meaning that anyone can and is welcome to edit almost anything and in particular, students can post notes, comments, pictures, whatever. Some rules, though -

• This wiki is a part of my (Dror's) academic web page. All postings on it must be class-related (or related to one of the other projects I'm involved with).
• I (Dror) will allow myself to exercise editorial control, when necessary.
• The titles of all pages and the names of all images related to this class should begin with "08-401/" or with "08-401-", just like the title of this page.

To edit, you must have a wiki account. To get one email a request to Dror at drorbn@math.toronto.edu, and include:

• Your first and last name.

Some further editing help is available at Help:Contents.

Marking Scheme

There will be one term test (25% of the total grade) and a final exam (50%), as well as about 10 homework assignments (25%).

The Term Test

The term test will take place in class on February 27. A student who misses the term test without providing a valid reason (for example, a doctor’s note) within one week of the test will receive a mark of 0 on the term test. There will be no make-up term test. If a student misses the term test for a valid reason, the weight of the problem sets will increase to 35% and the weight of the final exam to 65%.

Homework

Assignments will be posted on the course web page approximately on the weeks shown in the class timeline. Typically an "in preparation" version of any assignment will be posted a bit before class and the "in preparation" tag will be removed shortly after class, once our progress in class is precisely measured. Assignments will be due in class a week after they are assigned and they will be marked by the TA, usually within another week. All students (including those who join the course late) will receive a mark of 0 on each assignment not handed in; though to allow you some leeway, in computing the homework grade your worst two assignments will not count. I encourage you to discuss the assignments with other students or even browse the web, so long as you do at least some of the thinking on your own and you write up your own solutions. Remember that cheating is always possible and may increase your homework grade a bit. But it will hurt your exam grades a lot more.

Good Deeds

Students will be able to earn up to 25 "good deeds" points throughout the year for doing services to the class as a whole. There is no pre-set system for awarding these points, but the following will definitely count:

• Drawing a beautiful picture to illustrate a point discussed in class and posting it on this site.
• Taking class notes in nice handwriting, scanning them and posting them here.
• Typing up or formatting somebody else's class notes, correcting them or expanding them in any way.
• Writing an essay expanding on anything mentioned in class and posting it here; correcting or expanding somebody else's article.
• Doing anything on our 08-401/To do list.
• Any other service to the class as a whole.

Good deed points will count towards your final grade! If you got $n$ of those, they are solidly your and the formula for the final grade below will only be applied to the remaining $100-n$ points. So if you got 25 good deed points (say) and your final grade is 80, I will report your grade as $25+80(100-25)/100=85$. Yet you can get an overall 100 even without doing a single good deed.

Galois at the age of fifteen from the pencil of a classmate. He was young-looking for his age and had black hair.

Class Photo

To help me learn your names, I will take a class photo on the third week of classes. I will post the picture on the class' web site and you will be required to identify yourself on the Class Photo page of this wiki.

On Galois

The first paragraph of the Wikipedia entry on Galois:

Évariste Galois (IPA: [evaʁist gaˈlwa]; October 25, 1811 – May 31, 1832) was a French mathematician born in Bourg-la-Reine. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, thereby solving a long-standing problem. His work laid the foundations for Galois theory, a major branch of abstract algebra, and the subfield of Galois connections. He was the first to use the word "group" (French: groupe) as a technical term in mathematics to represent a group of permutations. A radical Republican during the monarchy of Louis Philippe in France, he died from wounds suffered in a duel under murky circumstances at the age of twenty.