MTH 1322 Rings, Fields, and Number Theory

Prof. A. Iarrobino

Spring 2002

Prerequisites

Math 1321 (Group theory), or permission of instructor

Course Description/Goals

We will begin by a review and extension of the group theory of Math 1321, including a study of symmetry and counting, and of group actions on a set. We then introduce rings and fields, and study field extensions and invariants of field extensions sucha as degree, and we introduce the Galois group Aut_F(E) of automorphisms of an extension E over a base field F. For example, Galois theory assigns the group {e} to the extension Q[real cube root of 2] over the rationals Q, and the symmetric group S_3 to the extension E= Q[all cube roots of 2] over the rationals Q). As conclusion of the course we begin the study of the main result of Galois theory, which assigns to each subgroup G of the Galois group Aut_F(E) the subfield extension G'=E^G left fixed by G. The basic result of field theory, and the concept of degree of an extension show the impossibility in the first two of the following three classical problems. The notions of Galois theory, and of constructible group are needed to resolve the third problem below.

A.It is impossible to duplicate the cube using ruler and compass. It is impossible to construct the cube root of two using a ruler and compass (two sticks of equal length, hinged at the top), given a unit length.
B. It is impossible to trisect an arbitrary given angle with the same tools.
C. It is imposssible to solve the general fifth degree polynomial equation f(x)=0, using a sequence of square, cube, ... , fifth, ... roots involving the coefficients. Thus, if ax^2+bx+c=0, one has a simple formula involving a square root for x, but no such formula is possible for ax^5+.....+f=0.

The Fundamental Theorem of Galois Theory states that for a good (= Galois) extension E of F, the fields K in between E and F correspond one to one with the subgroups of the Galois group of E over F. One goal of the course is that by the end, students will understand the above statement (see Theorem 32.1 in the text) and understand the steps in its proof. This is a beautiful theory, and exemplifies the aspect of mathematics in which apparently different objects are compared and found to be essentially the same.

There are applications of groups and Galois theory to many branches of mathematics such as coding theory, design of experiments, the topology of covering spaces, the geometry of algebraic curves, as well as to biology (via knot theory) and physics. This course has some flexibility, so that interest of students in specific topics can be reflected in the class.

Style

Style: We will focus on how we solve math problems including different methods and styles of working on problems. You will have a chance to present solutions in a supportive environment. What do mathematicians do? There will be an opportunity through optional essay and discussion to explore what it is like for us to solve math problems, to consider a career in or using mathematics. The course has three aspects:

A. Learning about algebra - Galois theory
B. Doing mathematics - problem solving, and
C. Reflection about the process.

One aim of the course is to help bridge the gap from earlier undergraduate mathematics courses toward the more sustained work of graduate study. One method will be an optional presentation.

Goals

Develop skill and self confidence in solving math problems.
Know when a problem is solved.
Learn the elements of Galois theory.
Understand some applications of fields and groups.
Develop the ability to present your work clearly to colleagues/peers.
Give an opportunity to reflect on doing mathematics.

Requirements

1. Homework problems from the text. There will be a chance to consult the class on problems you find difficult, and to redo problems if you'd like.

2. Work on certain designated problems, keeping a notebook record of how one has worked. These problems will be assigned a week ahead for discussion in class.

3. Participation in class - bring in questions, and volunteer when you are ready.

4. [Optional] Presentation of a topic of your choosing, with some elements beyond the text.

5. There will be a midterm and a final exam, and possibly a few quiizes.

Optional Presentation

About 30-45 minutes on a topic from Hungerford, Wickelgren,or outside reading. Proposals with references due May 4, Outlines due May 11. Ready to present by May 18.

Problem solving

Problems will be selected from the text, and from Hungerford, or ones I dream up. I will give feedback on your solutions. Grading will be on the number or problems solved, the difficulty of problems, quality of solutions, and your improvement. You will be able to rewrite-improve your solutions for credit, should you wish.

Final Exam

This will be take home, possibly with an optional in class portion during finals week. There will be about 80% problems, and 20% essay (choice of topics).

Grading

Problem solving/quizzes, possible midterm 3 units

Presentation, class participation 1 unit

Final exam 2 units Your course grade is the average of the top 5 units. Bonus of 2% for good attendance (see below).

Course Information

Instructor:	Professor Anthony Iarrobino, Office: 526 NI, phone: 373-5524; e-mail iarrobin@neu.edu
Time and Place:	MWF 10:30-11:35 at 509 Lake Hall, initial room.
Textbook: Joseph A. Gallian, Contemporary Abstract Algebra, 5th ed., Houghton Mifflin, 2002, ISBN 0-618-122141
Office hours:	Mon 9:15-10 AM, Mon 3-4 PM, Wed 12-1,3-4, Thurs. 3-4, or by appointment.
Math 1322 Class Notes	Brief notes about what was done in class, lists the HW assignments. Also, this may be a place to find samples and solutions of practice quizzes, sample exams, sample final, and answers to some of the exam questions given, if I don't pass out solutions.

See my teaching philosopy

Study groups: We may divide informally into study groups of two to four persons for work in class or some homework. These may be assigned, and if so will change from time to time. All exams are individual.

Attendance policy: In a small course, attendance is particularly important. Thus there will be a good attendance bonus of 2% for those missing zero-one classes. And a modest penalty of 0.5% per class missed > 4.

Assignments, specific info: See Math 1322 Class Notes for the actual assignments made, and links to online resources, references, in Galois Theory

Prof. Anthony Iarrobino
Department of Mathematics
Northeastern University
Boston, MA, 02115
Office:526 NI
Phone: (617) 373-5524
Email: iarrobin@neu.edu

Created: March 28,2002. Last modified: March 28,2002. URL:http://www.math.neu.edu/~iarrobino/AIMath1302.Spr02