Math 250A, Groups, Rings, and Fields
MWF 1-2 PM, 85 Evans
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About the course:
Professor: Elena Fuchs
Office: 851 Evans
Email: efuchs at math dot berkeley dot edu
Office hours: Mondays 2-4 and Wednesdays 2-3:30 or by appointment.
ANNOUNCEMENT: Our final will take place on Wednesday, Dec. 12, 7-10PM in 101 LSA (Life Sciences Addition). You are allowed to bring 8 pages of handwritten notes to the exam to use during the test. Please bring a blue book to record your solutions. We will be doing some review (and some category theory) during RRR week during regular class time in 85 Evans. We will have usual office hours both during RRR week and during the week of the final. Here are some problems which might help you prepare for the final. Some more practice problems can be found here. It is of course also a good idea to go over the homework assignments and midterm in preparation for the exam. Good luck!
This course is meant to equip the student with a thorough background in Algebra. This includes, very roughly, various standard important theorems on groups, rings, modules, and fields.
The assumption is that students taking this course have a solid background in undergraduate algebra -- the official prerequisites for this course are Math 113 and 114. Another assumption is that students will read about the basic notions in the text, and digest these notions, as well as those covered in class, through weekly homework assignments. There is a lot of material to cover, and so class time will be devoted only to main theorems and applications thereof: for example, definitions which you should either have seen in undergraduate algebra or could easily grasp by reading the book will not be covered during class time. On the other hand, we will cover some material in class that is either not emphasized in the book or deferred to an exercise.
To make it easier for you to prepare for the lectures, I will post in advance a weekly list of concepts we will cover in class, as well as notions that I will assume you know throughout each lecture on this website.
About the text:
The textbook for this class is Algebra by S. Lang, from which we will cover much of chapters I-VI. It is the standard book to use for a first year graduate algebra course, perhaps because it contains all of the topics one might cover in such a course and much more. It will serve you not only as a textbook for this course but also as an algebra reference throughout your mathematical endeavors.
Many students find that this book is much too dense. If you feel this way, I recommend that, rather than reading from a different book completely, you supplement your reading from Lang with additional reading from your undergraduate algebra book. Some other books I could recommend are
- Algebra by Artin
- Algebra by Birkhoff and MacLane
- Abstract Algebra by Dummit and Foote
- J.S. Milne's notes on group theory and Galois theory on this website
Keep in mind that Lang's book is written with a fairly mature mathematical audience in mind. The proofs often require the reader to fill in the details, and the examples sometimes assume material which you are by no means required to know (although these examples are usually very interesting). It is not expected that you will understand all of these examples.
Grades will be based on weekly homework assignments (50%), one take home midterm (15%), and one in class final (35%). Most homeworks (unless otherwise noted) will be due on Fridays in class or under my office door by 2PM. Collaboration on the homework is encouraged, but please do cite any sources which helped you to complete the assignment. No late homeworks will be accepted! The midterm will be handed out in class on Friday, October 12th, and due in class on Friday, October 19th. You may not collaborate with anyone on the midterm: any academic dishonesty on the exam will result in a score of 0 for the exam. Our final is scheduled for December 12, 7-10PM.
- 8/24: Isomorphism theorems, exact sequences, composition series. This week's suggestions on what to know before class are here.
- 8/27: Composititon series and Jordan-Holder theorem.
- 8/29: Solvable groups.
- 8/31: Solvable groups continued, beginning group actions.
- 9/5: Group actions and automorphism groups. This week's suggestions on what to know before class are here.
- 9/7: Semidirect products and applications.
- 9/10: Sylow Theorems.
- 9/12: Sylow Theorems and applications.
- 9/14: Beginning rings: ideals and Chinese Remainder Theorem. This week's suggestions on what to know before class are here.
- 9/17: Chinese Remainder Theorem, applications thereof, beginning proof that every PID is a UFD.
- 9/19: Continuing proof that every PID is a UFD.
- 9/21: If D is a UFD then D[x] is a UFD, beginning Hilbert's Basis theorem.
- 9/24: Finishing Hilbert's Basis theorem, beginning Mason-Stothers Theorem and ABC conjecture (see IV.7).
- 9/26: Finishing Mason-Stothers Theorem, beginning fields. This week's suggestions on what to know before class are here.
- 9/28: Transcendence degree continued.
- 10/1: Composite fields.
- 10/3: Splitting fields and algebraic closure.
- 10/5: Proof of existence of algebraic closure, starting separable extensions.
- 10/8: Separable extensions, finite fields.
- 10/10: Finishing off finite fields, beginning Galois theory. This week's suggestions on what to know before class are here.
- 10/12: L-valued characters, Galois extensions.
- 10/15: Galois groups and equivalent definitions of Galois extensions.
- 10/17: Fundamental theorem of Galois theory.
- 10/19: Finishing proof of fundamental theorem of Galois theory and applications.
- 10/22: Applications of Galois correspondence. This week's suggestions on what to know before class are here.
- 10/24: Fundamental theorem of algebra, starting cyclic extensions with solvability of polynomials in mind.
- 10/26: Radical extensions and Galois groups of composite extensions.
- 10/29: Galois' criterion for solvability of polynomials.
- 10/31: Solving cubics and quartics.
- 11/2: Solving quartics continued, some impossible and possible geometric constructions.
- 11/5: Galois groups of quartics, constructing n-gons. You might want to read the notes on geometric constructions posted above.
- 11/7: Finishing off geometric constructions, beginning modules. This week's suggestions on what to know before class are here.
- 11/9: Submodules, quotient modules, direct sums of modules, torsion modules.
- 11/14: Noetherian modules, cyclic modules.
- 11/16: Beginning discussion of finitely generated torsion modules over a PID.
- 11/19: Structure theorem for finitely generated torsion modules over a PID.
- 11/21: Free modules and structure theorem for finitely generated modules over a PID.
- 11/26: Proof of structure theorem for finitely generated modules over a PID, Jordan Canonical Form.
- 11/28: Projective modules vs. free modules.
- 11/30: Some category theory, abelian categories.
Some or all solutions will be posted on our course page on bspace once homeworks are handed in.
- Homework 1 due on Friday, 8/31.
- Homework 2 due on Friday, 9/7.
- Homework 3 due on Friday, 9/14.
- Homework 4 due on Friday, 9/21.
- Homework 5 due on Friday, 9/28.
- Homework 6 due on Friday, 10/5.
- Homework 7 due on Friday, 10/12.
- No homework due on Friday, 10/19. Instead the take-home midterm is due that day.
- Homework 8 due on Friday, 10/26.
- Homework 9 due on Friday, 11/2.
- Homework 10 due on Friday, 11/9.
- Homework 11 due on Friday, 11/16.
- Homework 12a due on Friday, 11/30 (there will be a 12b, so it might be a good idea to complete 12a by 11/26 or so).
- Homework 12b due on Friday, 11/30 (along with 12a)