MTH 619: Algebra
Fall 2019

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Instructor: Alexandru Chirvasitu

Lectures:

• MWF 10:00 - 10:50 AM, Math 235

Office: 216 Mathematics Building
Office hours: W 11-12
Email: achirvas AT buffalo.edu

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Textbook

• Abstract Algebra by Dummit and Foote

Other sources might come in handy for various topics we touch on. For commutative rings, for instance,

• Introduction to Commutative Algebra by Atiyah and Macdonald

is just about priceless (link opens a pdf).

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Reading

I'll probably ask you to read some of the sections in the textbook before most lectures, so we're in sync. I'll post the reading assignments here. Unless specified otherwise the sections are those in Dummit-Foote.

'Specified otherwise' might mean for instance AM for Atiyah-Macdonald (the other source given above).

The date is that of the lecture, so please do the reading before that.

	Due date	Assignment	Remarks
1	W Aug 28	7.1, 7.2
2	F Aug 30	7.3 Appendix II (category theory)	Category theory can have an enlightening effect on much of the math you learn and do, so you may as well get a leg up on the topic as soon as you can.
3	W Sep 4	7.4, 7.5, 7.6
4	F Sep 6	10.1
5	M Sep 9	10.2
6	F Sep 13	10.3
7	F Sep 20	10.4, 10.5
8	F Sep 27	13.1, 13.2
	F Oct 4	return 1st take-home exam
No classes the whole week of Oct 7
9	F Oct 18	13.4, 13.5
10	F Oct 25	13.6, 14.1
11	F Nov 1	14.2
No classes F Nov 8
12	F Nov 15	14.3, 14.4
No classes the whole week of Nov 25
13	F Dec 6	14.5, 14.6

Supplementary material

On occasion I'll post extra notes, comments, etc. in this space.

• (Co)limits in arbitrary categories unify a number of concepts you've seen before: direct sums and products, kernels and cokernels of morphisms between abelian groups or modules, etc. Here are some references on the topic.
  • The Stacks Project (this is a priceless resource for algebraic geometry and much more; consider this an excuse to get you acquainted with it)
  • nlab is another resource you'll find yourself using often for all things categorical.
  • The first three sections of this expository paper should be helpful.
• I promised I'd find a reference for the result saying that if L/K is an algebraic extension such that all polynomials in K[x] have roots in L then L is algebraically closed.
  This short paper proves that. Now that we've talked about (in)separability it'll make more sense, though I expect we'll have to cover some more ground for this to become fully comprehensible.

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Homework

All assignments are due in class on the dates indicated in the table below.

No late homework for any reason, but we are dropping the two lowest scores.

	Due date	Assignment	Remarks
1	F Sep 6	7.2: 3,4,5,6,7 7.3: 16, 20, 26, 28, 29, 34, 35 7.4: 11, 13, 33, 34, 37 7.5: 4
2	F Sep 13	10.2: 6, 9, 10 10.3: 2, 7, 9, 23
3	F Sep 20	10.4: 3, 4, 14 10.5: 4, 5, 7, 9, 18
4	F Sep 27	13.1: 3, 4, 5 13.2: 1, 4, 6, 7, 12
Exam 1	F Oct 4	10.4: 15 10.5: 16 13.2: 13, 16
5	F Oct 18	13.4: 2, 5, 6 13.5: 3, 4, 5, 7, 8
6	F Oct 25	13.6: 1, 6, 7, 10 14.1: 4, 5, 7, 9
7	F Nov 1	14.2: 7, 8, 9, 10 15, 17, 18, 19
8	F Nov 15	14.3: 3, 4, 8, 9 14.4: 1, 2, 6, 7
Exam 2	F Dec 6	14.5: 5, 10 14.6: 17, 24

The grader will assign each homework a grade out of 10 with only a few problems being graded (3 or 4 or so). There will also be points awarded for the overall completeness of the assignment. I won't be telling you in advance which problems are being graded though..

Do collaborate on the homework if you like, but write up your own solutions. I also strongly advise you to have a look at the UB Academic Integrity Policy, as it very much applies to this class.

And by all means drop by at office hours if you need a hand.

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Exams

We're having two take-home exams. The dates are as follows

• Exam 1: Fr Oct 4
• Exam 2: Fr Dec 6

Exam policy:

Those are the due dates. I will post the problem lists a few days before, so you will have plenty of time to work on them.

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Grading

• homework : 50%
• exams : 50%

We'll drop the two lowest homework scores.

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If you have any questions, don't hesitate to email me.

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