# 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, jarod@stanford.edu, office 380-J.

Problem sets: 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.

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, isomorphism, automorphism.
• 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, pdf).
• 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, pdf).
• 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. (practice midterm)
• Class 12 (M Nov. 1): problems.
• W Nov. 3: Midterm (ps, 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, pdf).
• 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, pdf).
• 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, pdf).
• 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: (ps, pdf).
• 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.