Algebra I Autumn 2018

Prof. Dr. Rahul Pandharipande
Johannes Schmitt
Wed 13-15 - HG E 5
Fri 08-10 - HG E 5
Exercise session
Mon 14-16 or Wed 15-17 - rooms: see below

Ferienpräsenz Spring 2019 During the semester break there are four events where you can ask questions concerning various classes (including Algebra I). You find the dates and a link to a doodle (where you should register!) at the website of Groups 1 and 4.

The main reference for the course is J. Rotman, "Advanced modern algebra, 3rd edition, part 1" (link only works from ETH computers and via vpn).
Below we give page references from this book for each week of the course.

We can also recommend the following two books for the course: For more literature as well as more information about the course, please see the VVZ page here.

Update: The midterm exam now takes place on Friday in the first week of the Spring semester (see below). It takes place in the usual slot of the lecture and instead the first lecture of the semester will happen on Monday, the 18th of February 2019, 8:00-10:00 in the usual lecture room (HG G5).

As described on the VVZ page of the course, there will be a voluntary midterm exam at the beginning of the Spring Semester 2019. This is a written exam that will take place on Friday, the 22nd of February 2019, 8:00-10:00 about the material from the lecture Algebra I. If your grade for the midterm exam is higher than your grade for the obligatory final exam in August, then the midterm exam will count for 20% of your course grade. If not, then the midterm exam will play no role in the course grade. The precise formula is
(course grade) = max((final exam grade), 0.8*(final exam grade) + 0.2*(midterm exam grade)).
Further details concerning rooms etc. will be given at a later point. The midterm exam from last year's course can be found here.

For those students taking a separate oral exam about the content of Algebra I, here is a list of questions to help you prepare for the exam. As part of the exam, we will ask you to present the solution of 2-3 exercises from this list on a blackboard, but there will also be other questions.

The new exercises will be posted here on Fridays. We expect you to look at the problems over the weekend and to prepare questions for the exercise classes on Monday/Wednesday.

Please hand in your solutions by the following Friday at 12:00 in your assistant's box in HG J68. Your solutions will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J68.

week material covered (page numbers: see main reference) assignment due by solutions
1 General information on the course, arithmetic with the integers, Zorn's lemma. (pages 9-13, 313-314, 319-320) A1 Sept 28 S1
2 Category Theory, first definitions on Rings (pages 441-445, 461, 29-37); see also this summary A2 Oct 05 S2
3 Fraction field, Polynomial Rings (pages 36-38, 41-45, 48-49) A3 Oct 12 S3
4 Ideals, First Isomorphism Theorem (pages 50-51, 55-60, 61: Ex A-3.52, 21: A-2.25, 74-75) A4 Oct 19 S4
5 Prime ideals and maximal ideals, Existence of maximal ideals, Arithmetic of Polynomials with coefficients in a field, Chinese Remainder Theorem, definition of UFD (pages 74-75, 315, 62-64, 104) A5 Oct 26 S5
6 UFDs (pages 104-111) A6 Nov 02 S6
7 First definitions on Groups, Group homomorphisms, Subgroups (pages 127-134, 150-155, 143-145) A7 Nov 09 S7
8 Conjugacy, Normal subgroups, quotient groups, Isomorphism theorems for groups (pages 153-155,159-165,141-143) A8 Nov 16 S8
9 Group action (see notes), Sylow theorems A9 Nov 23 S9
10 Sylow theorems (continuation, see notes from last week), the symmetric group (pages 116-126, 174-176; see also these notes) A10* Nov 30 S10
11 Field extensions, algebraic elements (pages 76-82) A11 Dec 07 S11
12 Splitting field, finite fields (pages 83-88) A12 Dec 14 S12
13 Algebraic closure, Finite fields, modules over a ring (pages 288-291, 344, 371-377, also this reference for algebraic closure) A13 No hand-in. S13
14 Finitely generated modules over a PID (pages 371-377) A14 No hand-in. S14

* corrected small mistake in Exercise 8

The exercise classes start in the second week of the semester. All exercise classes are held in English. Please enroll in one of the exercise classes using this link.

Mo 14-16LFW C 4Igor Balla
Mo 14-16ML F 39Younghan Bae
Mo 14-16ML J 37.1Yujie Wu
We 15-17CAB G 51Dominique Heyn
We 15-17LEE C 114Pascal Schilde
We 15-17ML H 34.3 Rajko Nenadov

For those of you interested in the research of Prof. Pandharipande, here is a link to a lecture given by him at the International Congress of Mathematicians (ICM) 2018 in Rio.