Groups, Rings, and Fields (Fall 2019)

Groups, Rings, and Fields (Fall 2019)

Location

Tuesday Thursday 9:30A-11:00A Dwinelle 219 (this might change). Office hours 11:00-12:30  Tuesday Thursday 927 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, 
2, 3 Sept 3, 5 Subgroups I.3-5 I 12, 13,  19, 20,  24,  26,
4, 5 Sep  10,12 Sylow theorems, abelian groups  I.6-10 I 30, 31, 32, 34, 35, 38, 41, 42
6, 7 Sep 17, 19 Categories, Free groups I.11-12 I 50, 51, 52, 53
8 Sep 24 Rings II.1-3 II 1, 8, 11, 12
9 Sep 26 Midterm 1 I (groups)
10, 11 Oct 1, 3 Localization, UFD, modules II.4-5, III.1-2 II 5, 9,13, 14  III 1, 3, 
12, 13 Oct 8, 10 Free modules III.3-6 III 5, 9, 10
14, 15 Oct 15, 17 Homological algebra III.7-10 III 14, 15, 17, 21,
16, 17 Oct 22, 24 Polynomials IV.1-6 IV 1, 3, 5, 7, 10, 
18, 19 Oct 29, 31 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)

 

 

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.