Groups, Rings, and Fields (Fall 2019)

Groups, Rings, and Fields (Fall 2019)

Location

Tuesday Thursday 9:30A-11:00A. The room  has changed to 3108 Etcheverry. Office hours 11:00-12:30  Tuesday Thursday 927 Evans Hall.  Ian  Gleason: Office hours Friday 2:00-4:00, 1093 Evans Hall.

Textbook

"Algebra" (3rd edition) by Serge Lang.  Click on the link from a UCB address to get a paper copy for $25 or a free electronic copy. We will cover chapters 1 to 6.  

Syllabus

 Group theory, including the Jordan-Holder theorem and the Sylow theorems. Basic theory of rings and their ideals. Unique factorization domains and principal ideal domains. Modules. Chain conditions. Fields, including fundamental theorem of Galois theory, theory of finite fields, and transcendence degree.

The lectures will emphasize examples and calculations over theorems or proofs.  Course philosophy: theorems will only be  proved in lectures when they are needed for an example. 

Prerequisites

Undergraduates who wish to take Math 250A are strongly advised to take Math 113 first, and  recommended to take Math 114.

Grading

Grading will be based on 2 midterms, the final, and homework.  There will be no makeup exams. If one exam is missed the grade for it will be estimated based on the other exams and the corresponding homework. The lowest two homework scores will be dropped.

Midterm  and final questions will mostly be closely based on problems in the textbook, especially homework problems, or examples done in lectures. You may bring one sheet of notes to the midterms and final.

Reading and homework

Homework is due on Tuesday the week after it is assigned. Homework will not be accepted after Dec 10. 

 

  The following table is provisional and may be changed. 

Lecture Date Topic Reading Homework
1 Aug 29 Groups I.1-2 I 1, 2, 9, 10, comments
2, 3 Sept 3, 5 Subgroups I.3-5 I 12, 13,  19, 20,  24,  26,comments
4, 5 Sep  10,12 Sylow theorems, abelian groups  I.6-10 I 30, 31, 32, 34, 35, 38, 41, 42, comments
6, 7 Sep 17, 19 Categories, Free groups I.11-12 I 50, 51, 52, 53 Comments
8 Sep 24 Rings II.1-3 II 1, 8, 11, 12  Comments
9 Sep 26 Midterm 1 I (groups)
10, 11 Oct 1, 3 Localization, UFD II.4-5, III.1-2 II 5, 9,13, 14  III 1, 3, Comments
12, 13 Oct 8, 10 Modules, tensor products III.3-6 III 5, 9, 10 Comments 7
14, 15 Oct 15, 17 Duality, limits, homology III.7-10 III 14, 15, 17, 21, Comments 8
16, 17 Oct 22, 24 Polynomials, Noetherian rings IV.1-6 IV 1, 3, 5, 7, 10, Comments 9
18, 19 Oct 29, 31 Symmetric functions, Resultants, power series IV.7-9 IV 13, 18, 25, 26, 27
20 Nov 5 Algebraic extensions V.1-4 V 1, 4, 6, 7, 9, 11, 19
21 Nov 7 Midterm II II, III, IV (rings)
22, 23 Nov 12, 14 Galois extensions V.5-6 VI.1-2 V 22, 23, 24  VI 1
24, 25 Nov 19, 21 Cyclic extensions VI.3-6 VI 7, 8, 13ab, 18, 19, 21
26 Nov  26 Solvable extensions VI.7-12 VI 23, 30, 31, 32, 33
27, 28 Dec 3, 5 Infinite extensions VI.13-15

VI 43, 44, 46

Dec 17, 3:00-6:00 Final V,  VI (fields)

Using TEX: I encourage students to write up their problem-set solutions in TEX, more specifically LATEX. This is a powerful mathematical typesetting program which is the standard way to write papers in math, physics, and computer science. Thus learning to use TEX is a valuable skill if you work in such fields.

The easiest way to use TeX is overleaf which works in any browser. 

You also can freely download versions of TEX onto your computer.
If you use Mac OS, you can find it at MacTex.
If you use Windows, you can find it at proTeXt

Several guides to using TEX are listed on the department's computer support web pages. One beginner's guide can be found at texman A more comprehensive guide can be found at this link

The best way to start learning TEX is not by trying to compose the long header, but rather by having a file that already has a header, and then gradually modifying that file as you learn how TEX works.
For such a TEX file with a header, click LATEX-sample . Make a copy to play with, once you have downloaded TEX onto your computer. At first, don't modify anything above "\begin{document}".

 

 

 

Links related to the course:

Course Summary:

Date Details
Public Domain This course content is offered under a Public Domain license. Content in this course can be considered under this license unless otherwise noted.