# Math 552: Modern Algebra II - Spring 2015

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## Instructor and General Information

 Instructor: Luís Finotti Office: Ayres Hall 251 Phone: 974-1321 (don't leave messages! -- e-mail (finotti at math dot utk dot edu) me if I don't answer!) e-mail: finotti at math dot utk dot edu Office Hours: MW 9-10 or by appointment.

 Textbook: D. Dummit and R. Foote, Abstract Algebra, 3rd edition, 2003, Wiley. (ERRATA!) Prerequisite: Math 551 or equivalent. Class: MWF 11:15-12:05 at Ayres Hall 112. (Section 1.) Exams: Midterm: 03/04 (Wed), Final: 04/30 (Thursday) from 10:15am to 12:15pm, in our regular classroom. Grade: Roughly: 30% for HW + 30% for the Midterm + 40% for the Final. Note the weight of the HWs! (Since this is a graduate course, there will be more leeway on these weights.)

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## Course Description and Information

### Course Content

This is the second course of the graduate sequence in Modern Algebra. After I finish a few topics in Ring Theory leftover from last semester, we will cover topics in Modules and Field/Galois Theory this semester.

The amount to be covered is again very large, and thus the pace of the class might be a bit fast. I will assume you still remember Groups and Rings, and have some familiarity with Vector Spaces and Fields. For the latter two, I will only assume that you know basic topics that anyone should have seen in an undergraduate algebra course, or mentioned last semester. I might quickly remind you of some of these basic facts, but I might skip some altogether. Please, slow me down if I'm going too fast.

### Chapters and Topics

We should cover sections 9.3-6 (no Gröbner Basis), 15.1 (Noetherian rings only), 15.2 (Radicals only), 16.2, 10.1-10.5 (Projective modules only), 12.1-3. For Field and Galois Theory I will switch to Lang's Book. We will cover sections V.1-6 and VI.1-7 from this text, which roughly corresponds to sections 13.1-6 and 14.1-8 from Dummit and Foote. The HWs will be from Dummit and Foote, so you do not need to buy Lang's.

### Homework Policy

Homeworks will be posted regularly at the section Homework of this page. No paper copy of the HW assignments will be distributed in class. It is your responsibility to check this page often!

The HWs will be collected on Wednesdays. Each HW will have problems from the previous week (Monday, Wednesday and Friday lectures). The problems to be turned in, as well as due dates, will be clearly posted here. Note that not all of the problems turned in will be graded, but you won't know which until you get them back.

Problems likely to be assigned are posted below, but are subject to change. So, you can always start early, even if the assignment is not posted. (The list is likely incomplete, but chances of changing an assigned problem are small.)

Note that I might sometimes get too ambitious in posting problems, i.e., I might think we will cover a section during the week, put exercises from it in the next assignment, and then end up not being able to finish it. In this case I might have to take a few problems off the assignment. The bottom line is the following: the assignment is not final until I remove the "More to come" from it. (If you've done problems which were removed, just saved them for the following week.)

Finally, if there is still a "More to come" in an assignment on a Friday, please right away so that I can update it. If I delay in replying, you can proceed with the Problems Likely To Be Assigned.

No late HWs will be accepted, except in extraordinary circumstances which are properly documented.

It is your responsibility to keep all your graded HWs and Midterms! It is very important to have them in case there is any problem with your grade.

In my opinion, doing the HW is one of the most important parts of the learning process, so the weight for them is greater than the weight of a single midterm, and I will assume that you will work very hard on them.

Also, you should try to come to my office hours if you are having difficulties with the course. I will do my best to help you. Please try to come during my scheduled office hours, but feel free to make an appointment if that would be impossible.

### E-Mail Policy

I will assume you check your e-mail at least once a day. I will use your e-mail (given to me by the registrar's office) to make announcements. (If that is not your preferred address, please make sure to forward your university e-mail to it!) I will assume that any message that I sent via e-mail will be read in less than twenty four hours, and it will be considered an official communication.

### Blackboard and Feedback

There is a link for (general) Feedback on Blackboard. That is in fact a discussion board and I'd ask you to subscribe to it to receive e-mail notifications for posts. Please, post all comments and suggestions there as often as you want. (I really appreciate your input.) These can be posted anonymously (or not). Just make sure to check the option before posting! Others students and myself will be able to respond and comment. If you prefer to keep the conversation private (between us), you can send me an , but then, of course, it won't be anonymous.

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Here are some other books you might find helpful:

• S. Lang. Algebra'', 3rd Edition. Springer, 2005. -- Probably the best reference algebra book there is.
• I. Isaacs, Algebra: A Graduate Course'', 1st Ed., 1994. AMS (Originally published by Brooks Cole/Cengage Learning). -- Particularly good in group theory and non-commutative algebra.
• B.L. van der Waerden, Algebra I, II'', 2nd Ed., 2003, Springer. -- A classic.
• N. Jacobson, Basic Algebra I and II'', 2nd Ed.,
1. W H Freeman & Co. (Out of print.) -- Another standard book.
• T. Hungerford, Algebra'', 1st Ed., 1974, Springer. -- Another standard book.

Here are some which are more on the level of undergraduate algebra:

The first two books are considered easier'' books. The Artin's book is of a bit higher level (and has a slightly different focus).

The last one is a standard'' text for a first course in abstract algebra, but have a higher level of difficulty than the previous two. It's been used for the honors section of the undergraduate algebra course here at UT, and it might be even on the level of a graduate course in some parts.

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### Conduct

All students should be familiar and maintain their Academic Integrity: from Hilltopics, pg. 46:

The university expects that all academic work will provide an honest reflection of the knowledge and abilities of both students and faculty. Cheating, plagiarism, fabrication of data, providing unauthorized help, and other acts of academic dishonesty are abhorrent to the purposes for which the university exists. In support of its commitment to academic integrity, the university has adopted an Honor Statement.

All students should follow the Honor Statement: from Hilltopics, pg. 16:

Honor Statement

"An essential feature of The University of Tennessee is a commitment to maintaining an atmosphere of intellectual integrity and academic honesty. As a student of the University, I pledge that I will neither knowingly give nor receive any inappropriate assistance in academic work, thus affirming my own personal commitment to honor and integrity."

You should also be familiar with the Classroom Behavior Expectations.

We are in a honor system in this course!

### Disabilities

Students with disabilities that need special accommodations should contact the Office of Disability Services and bring me the appropriate letter/forms.

### Sexual Harassment and Discrimination

For Sexual Harassment and Discrimination information, please visit the Office of Equity and Diversity.

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## Problems Likely To Be Assigned

This list is subject to change without prior notice. The official assignments will be posted below.

Section 9.3: 1, 2, 4(a-d).

Section 9.4: 3, 4, 12. Also look at the easy/computational ones, such as 1, 2, 9, 10, 11, 18.

Section 9.5: 7. Take at look at the others.

Section 9.6: 1.

Section 15.1: 1, 2, 4.

Section 16.1: 2, 3, 4.

Section 16.2: 1, 3, 4.

Section 10.1:. (Most of these are quite quick and easy. At least take a look at them.) 2, 3, 4, 5, 7, 8, 13, 15, 18, 19, 20, 21, 23.

Section 10.2: 4, 6, 8, 9, 10, 11, 12, 13.

Section 10.3: Look at all of them, and do a few. There are too many problems that show nice (and easy) properties of modules. 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 22, 23, 24(a)-(e).

Section 10.4: 3, 4, 5, 6, 7, 9 (use 8(c) without proving it), 10, 11, 15, 17, 18, 20, 24, 25.

Section 10.5: 1, 3, 4, 6, 7, 8, 9, 10, 11, 12 13, 14(a)-(b).

Section 12.1: 2, 3, 4, 6, 8, 9, 15, 21, 22. (Exercises 16 to 19 are important to justify the algorithm for rational canonical form.)

Section 12.2: 1, 2, 3, 4, 6, 7, 9, 10, 13, 17, 18, 19, 20, 21. (Exercises 22 to 25 are important to justify the algorithm for rational canonical form.)

Section 12.3: 2, 10, 17, 19, 22, 25, 26, 29, 33. Also make sure you do a few computational ones.

Section 13.1: 2, 4, 8.

Section 13.2: 1, 4, 8, 13, 15, 17, 18, 19, 20, 22.

Section 13.4: 1, 2, 3, 4.

Section 13.5: 2, 3, 4, 5, 6, 8, 9, 10.

Section 14.9: 1, 2, 3.

Section 14.3: (You don't need Galois Theory here, but you can use it if you want.) 3, 4, 5 (this statement is not so good -- identity is an isomorphism -- so try to show that the roots of the second polynomial are in the splitting field of the first), 6, 7, 10, 11.

Section 14.1: 1, 2, 3, 4, 5, 6, 7.

Section 14.2: 1, 3, 4, 6 (look at the computations on pg. 557), 7, 8, 9, 11, 13, 14, 15, 16..

Section 14.4: 1, 2, 3, 4 (the hint suggests that f is separable; the statement is true in general, and so do it for the general case; the hint seems to go bad in this situation, so maybe you shouldn't try to follow it; finally, note that for the book, Galois implies finite, so assume that K/F is finite), 5(a)-(b), 6 (note that this is the hard way; it's easier to show explicitly that it is not simple), 7, 8 (I think it needs a little of flat modules).

Section 13.6: 1, 2, 3, 5, 6.

Section 14.5: 3, 5, 6, 7, 8, 9, 10, 11, 12.

Section 14.6: 2(b), (c), 4, 5, 10, 18, 20, 28.

Section 14.7: 3, 4, 5, 6, 7, 8, 12, 13.

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## Homework

HW1 - Due on Wednesday 01/14:

Section 9.3: Turn in: 2, 4(a-d). Do not turn in: 1.

Section 9.4: Turn in: 4, 12. Do not turn in: 1, 2, 3, 9, 10, 11, 18.

HW2 - Due on Wednesday 01/21:

Section 9.5: More to come!

PLEASE, HIT "REFRESH" (OR "RELOAD") IN YOUR BROWSER WHEN VISITING THIS PAGE!!!!!!! I usually get messages asking for the update in the HW when it has already been updated. Since I change this page often, some times the browser don't see the changes. But, if you hit refresh and there is still problems missing, feel free to write me.

If it is already Friday afternoon and there still is a "More to come" after the HW assignment due on the coming Wednesday, write me an e-mail at , and I'll update it and let you know.

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