Algebra I (Spring 2018)
University of Pittsburgh

Description
We will cover material related to groups, fields, and Galois theory. Some other optional related material will be covered as time permits.

Prerequisites: Undergraduate level abstract algebra (such as MATH0430)
Announcements
o April 17: We are finished with splitting fields, separable extensions and existence and uniqueness of finite fields (Sec. 13.4, 13.5). We have started with Galois theory (sec. 14.1). We will prove Fundamental Theorem of Galois Theory (Sec. 14.2) on Wednesday and Friday.
o April 17: I have posted the last homework. Since not much time left before the final, the last homework is optional (considered as a bonus). It is due Wednesday next week (April 25).
o April 17: There are more practice problems from DF posted (below). There will not be more practice problems.
o April 17: Final exam is on Friday April 27 starting at 12pm (I am thinking of having 3 hours time for the exam).
o March 28: Practice problems from DF will be posted very soon.
o March 28: Some bonus project topics posted. See below in homework section.
o March 28: HW 9 posted.
o March 28: Past few lectures we have started with field theory. We reviewed fields, rings and ideals (basically a quick summary of DF Sec. 7.1-7.4 and 9.1-9.3). We are now covering Sec. 13.1 and 13.2. These two sections are essential for the rest of the course.
o March 28: We have finished group theory. The last topic was semi-direct product of groups (Sec. 5.5). We skipped 4.6 (Simplicity of A_n) as well as 5.1-5.4.
o March. 8: Here are midterm solutions (thanks to Shaoyu).

o Feb. 28: Here are midterm problems. I will write solutions to midterm later.

o Info about midterm:
- It covers 1.1-1.7, 2.1-2.5, 3.1-3.5, 4.1-4.2
-You need to know all the homework problems.
- There will be 5-6 questions, with one question about definitions/statement of theorems (e.g. Definition of center of a group or statement of First Iso Theorem or Lagrange's Theorem) and one question proof of a theorem from class or the text (DF) (e.g. prove Cayley's Theorem).
o Midterm Wednesday February 28 in class (50 min).
o Xiao’s office hours are in Posvar Hall 1200A (Computer Lab). Her schedule: is Monday 11-1, Wednesday 3-4 and Friday 1-2.

o Jan. 10: The TA for the course is Xiao (Nova) Chang. Please talk to her about anything related to the grading and homeworks.

o Jan. 10: First homework posted (see below). From now homeworks will not be announced just look for them below every week around Wednesday. Each homework is due the week after in class on Wednesday.

o Jan. 8: First class.
Homeworks and other online material
Optional bonus project topics:
- Simplicity of A_n and PSL(F_p). Reference for A_n: most texts in groups theory such as DF and Fraleigh. Reference for PSL(F_p): Paul Monsky, "Frobenius’ Result on Simple Groups of Order (p3 − p)/2" The American Mathematical Monthly, Vol. 120, No. 8 (October 2013), pp. 725–732, or any other reference you can find.
- Rubik's cube. A reference: "Group Theory and the Rubik’s Cube" by Janet Chen. In there it proves a criterion for what configurations in Rubik's cube are valid (last section).
- Fundamental group of a topological space. Reference: any book in algebraic topology. Give definition of fundamental group and compute (using standard theorems like Van Kampen) some spaces such as union of circles meeting at one point and a torus.
- Geometric constructions, origamy and field extensions: A reference: David Cox "Galois theory" (Section 10.3). Also Geretschläger "Euclidean constructions and the geometry of origami". Math. Mag. 68 (1995), no. 5, 357–371. You can find other references by googling.

I expect 2-3 pages (or more no restriction on number of pages!) typeset in LaTeX about one of the above topics containing at least one non-trivial theorem with proof (or a detailed sketch of proof). No need to say that you should understand the proof(s) well! (of course you can take it from any source you want). You have until the last day of classes.

HW11 (due Wednesday April 25th in class, this homework is optional or bonus):
DF Sec. 14.1: 1, 5, 8
DF Sec. 14.2: 10, 16

HW10 (due Friday April 13th in class):
DF Sec. 13.3: 4, 5
DF Sec. 13.4: 2, 3, 5
DF Sec. 13.5: 4

HW9 (due Wed. April 4th in class):
DF Sec. 13.1: 1, 2, 5
DF Sec. 13.2: 1, 4, 7, 10, 14, 21
- Problem: Construct a finite field of order 9 as a quotient of the polynomial ring F_3[x] where F_3 is the field (Z_p, + mod 3, x mod 3).

HW8 (due Fri. March 23th in class):
DF Sec. 4.5: 9, 36, 40
DF Sec. 5.5: 1, 2, 16

HW7 (due Wed. March 14th in class):
DF Sec. 4.1: 9
DF Sec. 4.2: 3(a), 7, 8
DF Sec. 4.3: 12, 13
DF Sec. 4.4: 3

HW6 (due Fri. Feb. 23th in class):
DF Sec. 3.4: 7, 8
DF Sec. 3.5: 11, 12
DF Sec. 4.1: 4
- Problem: Let a group G act on a set X. Take x in X and let G_x be its stabilizer subgroup and let H=G.x be the G-orbit of x in X. (1) Prove Orbit-Stabilizer theorem i.e. the map g.x goes to gH gives a one-to-one correspondence between the orbit G.x and the quotient G/H. (2) Let y=g.x be another element of in the orbit G.x. Show that the stabilizer subgroup G_y is conjugate to G_x. (see also Sec. 4.1 #2)

HW5 (due Fri. Feb. 16th in class):
DF Sec. 3.2: Problems 14, 16
DF Sec. 3.3: Problems 3, 7
-Problem: For any natural number n show that the sum of Phi(d), where d runs over all divisors of n, is equal to n itself. Here Phi is the Euler Phi function. Hint: consider a cyclic group G of order n and count the number of generators for all subgroups of G.
-Problem: Show that the converse of Lagrange’s theorem is false. That is, there is a group G and a divisor d | |G| such that G has no subgroup of order d. Hint: take G = alternating group A_4.
-Problem: (Lagrange) Let f be a polynomial in n variables. Show that the number of permutations in S_n that fixes f divides n!.
-Problem: Prove that K is normal subgroup of G if and only if the equivalence relation (on G) defined by (left cosets of) K respects the group operation.

HW4 (due Wed. Feb. 7th in class):
DF Sec. 2.4: Problems 9, 17 (Zorn’s lemma appears in Appendix 1)
DF Sec. 3.1: Problems 1, 12
-Problem: Let F be any field. Show that there is a 1-1 homomorphism from S_n to GL(n,F) given by the action of S_n on the indices i in the standard basis e_1,…,e_n. Combine this with Cayley’s theorem (page 120) to show that for every finite group G, there exists n, such that G is isomorphic to a subgroup of GL(n,F).
-Problem: Prove the statement from class about conjugacy classes in the symmetric group S_n. That is, two permutations are in the same conjugacy class if and only if the length of cycles in their cycle decompositions are the same.

HW3 (due Fri. Feb. 2nd in class):
DF Sec. 1.7: Problems 17, 18, 21
DF Sec. 2.1: Problem 11
DF Sec. 2.2: Problems 4, 14
DF Sec. 2.3: Problems 13, 26

HW2 (due Wed. Jan. 24th in class):
DF Sec. 1.3: Problems 13, 15
DF Sec. 1.4: Problems 7, 8
DF Sec. 1.5: Problems 3
DF Sec. 1.6: Problems 4, 14, 17, 24

HW1 (due Wed. Jan. 17th in class):
DF Sec. 1.1: Problems 7, 9, 31
DF Sec. 1.2: Problems 3, 11, 16

Some suggested practice problems from DF:

Sec. 1.1: 25, 31, 33, 35, 36
Sec. 1.2: 2-7, 9-17
Sec. 1.3: 2, 11, 13, 16, 18
Sec. 1.4: 10
Sec. 1.5: 1, 3
Sec. 1.6: 6, 18, 23, 24
Sec. 2.1: 8, 10, 14
Sec. 2.2: 5, 7, 13
Sec. 2.3: 3, 25, 26
Sec. 2.4: 15, 17, 19
Sec. 3.1: 1, 3, 7, 9, 14, 20, 22, 34, 36, 41
Sec. 3.2: 11, 14, 19, 22
Sec. 3.3: 1, 3, 6, 7
Sec. 3.4: 4, 8
Sec. 3.5: 2, 4, 8, 14
Sec. 4.1: 2, 3, 6, 10
Sec. 4.2: 1, 2, 3, 4, 8, 9, 10, 14
Sec. 4.3: 2-13, 20, 22, 25, 31, 34
Sec. 4.4: 1, 3, 5, 6, 9, 18
Sec. 4.5: 1, 4, 5, 6, 8, 10, 11, 18
Sec. 5.5: 1, 2, 7(c), 8
Sec. 13.1: 1-8
Sec. 13.2: 1-15, 17, 19-21
Sec. 13.3: 1, 4, 5
Sec. 13.4: 1-6
Sec. 13.5: 2, 4, 8
Sec. 14.1: 1, 2, 3, 4, 5, 6, 8, 9, 10
Sec. 14.2: 1, 2, 3, 4, 5, 7, 8, 12, 13, 14, 15, 17, 31
Sec. 14.3: 1, 3, 5, 6, 9

Course Information
o Time: MWF 12:00PM - 12:50PM

o Location: Thackeray 525

o There will weekly homework, a midterm and a final exam.

o Office hours: 11AM-12PM MWF or by appointment.

o Text: Dummit and Foote “Abstract Algebra”

o Grader: Xiao (Nova) Chang

Instructor's Information
Name: Kiumars Kaveh
Email: kavehk AT pitt.edu
Office: Thackeray #424
Office phone: 412-624-8331