Math 210A: Modern Algebra
Lectures: Mondays and Wednesdays 12:45-2:05 in 380-F
(not the time listed in the course guide).
Office hours: Mondays and Wednesdays
2:05-3 in 383-M (third floor of the math building).
I will have bonus office hours before the midterm and final.
Textbook: Lang's Algebra (revised third edition).
7-8 weekly problem sets 40%.
In-class mid-term 20%.
Final exam 40%.
Course assistant: Jarod Alper,
They will be mostly due on Fridays at noon, at Jarod
Alper's door (380-J). There will be an envelope there.
Lates will not be allowed, but the lowest score will be dropped.
This is the first course in a three-part sequence. From the course
guide: "Groups, rings, and fields, introduction to Galois theory.
Prerequisite: 120 or equivalent." The course will assume that you've
already had reasonable exposure to groups, rings, and fields.
For quals information, click here.
The course so far:
Problem sets in ps and pdf formats are below.
(Please let me know if you have trouble with the pdf version, or
if you want the dvi version.)
Class 1 (M Sept. 27): monoid, group, commutative/abelian, subgroup,
order, Cn, Sn, GxH, generators, group homomorphism (=morphism), ker, im,
Class 2 (W Sept. 29): normal, center, quotient=factor group, isomorphism
theorems part 1.
Class 3 (F Oct. 1): isomorphisms cont'd, correspondence theorem,
lattice of subgroups, tower/filtration of a group, normal tower,
abelian tower, solvable,
Jordan-Holder theorem take 1.
Class 4 (M Oct. 4): Zassenhaus (butterfly) lemma, Schreier's theorem,
proof of Jordan-Holder, action of group on a set, G-action, orbit Gi, Cauchy's
Theorem, isotropy group.
Problem sets 1 and 2 out (ps,
Class 5 (W Oct. 6, taught by Greg Brumfiel): more group actions,
the symmetric and alternating groups.
F Oct. 8: problem set 1 due.
Class 6 (M Oct. 11): alternating groups, intro to Sylow theorems.
Class 7 (W Oct. 13):
Problem set 3 out (ps,
F Oct. 15: Problem
set 4 out (ps,
pdf). Problem set 2 due.
Class 8 (M Oct. 18): direct sums and products of group,
free abelian groups, basis, rank.
Class 9 (W Oct. 20): torsion abelian groups; toward
the classification of finite abelian groups.
Class 10 (M Oct. 25): completing the classification,
introduction to semidirect products.
Problem set 3 due.
Class 11 (W Oct. 27): Semidirect products.
F Oct. 29: Problem set 4 due.
Class 12 (M Nov. 1): problems.
W Nov. 3: Midterm
pdf). Bonus office hours: Monday 2:05-4:30 (with 15 minute
break in middle), Tuesday 10 am to noon. My usual office hours on Wednesday will be canceled. Problem set 5 out (ps,
Class 13 (M Nov. 8): introduction to rings: commutative ring, identity, ring homomorphisms and isomorphisms, ideal, principal ideal domain, quotient ring, isomorphism theorems for rings.
Problem set 6 out (ps,
pdf). Office hours 2:05-4.
Class 14 (W Nov. 10, taught by Dan Bump): prime and maximal ideals,
fields, Zorn's lemma, multiplicative set, localization, local ring.
F Nov. 12: Problem set 5 due.
Class 15 (M Nov. 15): Chinese remainder theorem, irreducibles,
primes, greatest common divisor, unique factorization domain.
Class 16 (W Nov. 17): PID implies UFD, Noetherian ring, euclidean domain
implies PID, Z[i] is ED, Fermat's 2-squares theorem.
F Nov. 19: Problem set 6 due.
Problem set 7 out (ps,
Class 17 (M Nov. 22): Gauss' Theorem (R UFD implies R[x] UFD), Gauss' lemma(s), Eisenstein criterion, Luca's theorem.
Class 18 (W Nov. 24): fields, characteristic, extension, degree (E/F),
splitting field (uniqueness up to isomorphism).
F Nov. 26: Problem set 8 out (ps,
Class 19 (M Nov. 29): algebraic and transcendental extensions, existence
and uniqueness of finite fields of prime power order, independence of
set of mutually distinct characters, fixed field of set of automorphisms. Problem set 7 due.
Class 20 (W Dec. 1): impossibility of straightedge and compass construction of trisection of the angle, doubling the cube, and squaring the circle .
F Dec. 3: Problem set 8 due.
Practice final out:
W Dec. 8: Final exam 8:30 - 11:30 in 380-W.
I will have bonus office hours beforehand:
Sunday 2-4 pm and Monday and Tuesday 8-10 pm.
To my home page.
Department of Mathematics Rm. 383-M
Phone: 650-723-7850 (but e-mail is better)