Topics

I. Group theory

• Composition series and the Jordan-Hölder theorem in the context of groups with operators; simple groups and modules; solvable and nilpotent groups

• Group actions

• p-Groups and Sylow theorems

• Free groups; generators and relations

II. Rings, fields, modules

• Maximal and prime ideals

• Comaximal ideals and Chinese Remainder Theorem

• Noetherian rings

• Principal ideal domains and unique factorization domains

• Polynomial rings; Hilbert's Basis Theorem; Gauss's Lemma

• Finite, algebraic, and primitive field extensions

• Presentations of modules; structure of finitely-generated modules over principal ideal domains

III. Introduction to algebraic geometry

• Algebraic sets and varieties

• Hilbert's Nullstellensatz

• Nilpotent elements and radical

IV. Multilinear Algebra

• Tensor product of modules

• Tensor algebra of a bimodule

• Exterior algebra of a module over a commutative ring

MATH 6310 is the first semester of a two-semester basic graduate algebra sequence. The main topics to be covered in the second semester, MATH 6320, are Galois theory, representation theory of groups and associative algebras, and an introduction to homological algebra.

Prerequisites

The content of a solid undergraduate course in abstract algebra, comparable to MATH 4340. Students should know the basic definitions and properties of groups, rings, modules, and homomorphisms; substructures and quotient structures; isomorphism theorems; integral domains and their fraction fields. Very little of this material will be reviewed during the course.

Midterm 10/11/12 in class, covering through §6.3

Final 7:00pm–9:30pm, 12/6/12, MLT 207, covering everything

Exams will be taken without the aid of any textbooks or notes.

Calculators will NOT be allowed for the exams. Cell phones, MP3 players and other electronic devices (aside from watches) will NOT be allowed in the exam room, not even as time–keeping devices.

Accommodations may be made in the event of illness, a clash with another examinations, a conflict with religious observances, or other serious circumstances — please contact your instructor as early as possible. Students who miss exams and then present excuses that could have been given in advance will not be looked upon favorably. Do not make travel plans that conflict with the exams.

If you have been approved for extended–time or special–condition exams by Cornell Student Disability Services, notify your instructor in the first two weeks of the semester.

As you probably all know, homework is the most important part of this course since mathematics is really learnt by doing it. Accordingly homework will contribute a substantial portion of the grade.

Assignments will be due in class on Tuesdays. Papers should be stapled and your name clearly indicated. Late homework will not be accepted, as I will endeavor to make the assignments closely follow the classes. If you have a medical excuse or other serious circumstances, allowences will be made; contact me at the earliest opportunity. The lowest two scores across the semester will be dropped as everyone has a bad week sometime, and seniors and graduate students (which I think almost all of you are) sometimes have other pressing one-off commitments.

Due to constraints on resources, not all problems may be graded.

Collaborating. You may collaborate with other students on homework. I believe, however, that for maximum benefit, you should try hard to do all the problems yourself before consulting others. What you turn in should represent your own solutions expressed in your own words, even if you arrived at these solutions with others. Remember, you are doing the homework to learn the material; do not try to defeat its purpose. Copying someone else's homework or copying solutions from the internet or elsewhere and presenting it as your own defeats the purpose of homework (and violates Cornell's Academic Integrity Code).

Writing well. Please organise your answers thoughtfully, neatly and legibly. Don't turn in your first draft. Pay attention to spelling, grammar, and punctuation. Proofs should be written clearly and completely, using good English, complete sentences, and adequate detail. Imagine your fellow-students as your readers — ask yourself whether they would they be able to follow your arguments. You may be surprised to discover how much you learn in the process of writing homework solutions carefully. I think a great deal of consolidation of your knowledge takes place in the process. Please take this aspect of the homework seriously.

You may find it useful to write your solutions in LaTeX. It will be time consuming at first (if you don't already know LaTeX), but you will end up with a valuable skill that you will need eventually. Here's a guide to getting started. Well hand-written homework is fine instead.