Groups, Rings, and Fields (Fall 2017)
Location
Tuesday Thursday 12:30-2:00, 247 Cory. Office hours 11:00-12:30 Tuesday Thursday 927 Evans Hall. The GSI is Justin Chen. His office hours are in 959 Evans Hall and will be held 2:00-3:00 Tuesdays, and 2:00-4:00 Wednesdays.
Textbook
"Algebra" (3rd edition) by Serge Lang. Click on the link to get a paper copy for $25 or a free electronic copy. We will cover chapters 1 to 6.
Catalog Description
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.
Prerequisites
Undergraduates who wish to take Math 250A are advised to take Math 113 and Math 114 first.
Grading
Grading will be based on 2 midterms, the final, and homework. The lowest of the 3 exam scores will be dropped. There will be no makeup midterms or finals. The lowest three 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.
Daniel Raban has some notes for the lectures here
The following table is provisional and may be changed.
Lecture | Date | Topic | Reading | Homework |
1 | Aug 24 | Groups | I.1-2 | 1, 2, 9, 10, |
2, 3 | Aug 29, 31 | Subgroups | I.3-5 | 12, 13, 19, 20, 24, 26, |
4, 5 | Sep 5, 7 | Sylow, abelian | I.6-10 | 30, 31, 32, 34, 35, 38, 41, 42 |
6, 7 | Sep 12, 14 | Categories, Free groups | I.11-12 | 50, 51, 52, 53 |
8 | Sep 19, | Rings | II.1-3 | 1, 8, 11, 12 |
9 | Sep 21 | Midterm 1 | I (groups) | |
10, 11 | Sep 26, 28 | Localization, UFD, modules | II.4-5, III.1-2 | II 5, 9,13, 14 III 1, 3, |
12, 13 | Oct 3, 5 | Free modules | III.3-6 | 5, 9, 10 |
14, 15 | Oct 10, 12 | Homological algebra | III.7-10 | 14, 15, 17, 21, |
16, 17 | Oct 17, 19 | Polynomials | IV.1-6 | 1, 3, 5, 7, 10, |
18, 19 | Oct 24, 26 | Resultants, power series | IV.7-9 | 13, 18, 25, 26, 27 |
20 | Oct 31 | Algebraic extensions | V.1-4 | 1, 4, 6, 7, 9, 11, 19 |
21 | Nov 2 | Midterm II | II, III, IV (rings) | |
22, 23 | Nov 7, 9 | Galois extensions | V.5-6 VI.1-2 | V 23, 24 VI 1 |
24, 25 | Nov 14, 16 | Cyclic extensions | VI.3-6 | 7, 8, 13ab, 18, 19, 21 |
26 | Nov 21 | Solvable extensions | VI.7-12 | 23, 29, 30, 31, 32 |
27, 28 | Nov 28, 30 | Infinite extensions | VI.13-15 |
43, 44 |
Dec 14 3:00-6:00 | Final | I to VI |
Links related to the course:
- Mathematicians: Artin, Cayley, Eisenstein, Euclid, Galois, Grothendieck, Hilbert, Kummer, Lang, Emmy Noether, Serre, Sylvester. Van_der_Waerden .
Course Summary:
Date | Details | Due |
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