- Monday 3/20, Hales, review of rings, ideals, fields, and extensions
- Wednesday 3/22, Fedorov, D&F Section 13.1, basic theory of field extensions
- Friday 3/24, Fedorov, D&F Section 13.2, algebraic extensions
- Monday 3/27, Xu, D&F, Section 13.3, classical straightedge and compass constructions
- Wednesday 3/29, Yarosh, D&F, Section 13.4 splitting fields
- Friday 3/29, Yarosh, D&F, Section 13.4, algebraic closures
Perform the following calculations in GAP. You can use Sage Cell and choose GAP in the language menu at the bottom-right-hand corner of the input cell. You can cut and paste your answers into the submitted homework.
- compute 77+31;
- compute the set of divisors of 60: DivisorsInt(60);
- compute the greatest common divisor Gcd(33,770);
- compute 17 mod 3;
- compute a and b such that a*64+b*33 = 1, using GcdRepresentation(64,33); check your answer in gap
- find Factors(2^126-1);
- check the largest factor is prime: IsPrime(77158673929);
- compute the square (a^2;) of the permutation a:=(1,2,3,4)(6,5,7);
- compute the “left to right” (warning!) product of the cycles a:= (1,2,3); b:=(2,3); a*b;
- make a list of representatives of the permutation group generated by (1,2,3,4,5) and (2,5)(3,4): g:= Group((1,2,3,4,5),(2,5)(3,4)); c:= ConjugacyClasses(g);
- Calculate the conjugacy classes of A5: g:= AlternatingGroup(5);
- Find the derived subgroup of A4: DerivedSubgroup(AlternatingGroup(4));
- Calculate the 2-Sylow subgroup of S4: SylowSubgroup(SymmetricGroup(4),2);
- Show that the subgroup of S4 generated by (1,2,3) and (2,3,4) is A4: StructureDescription(Group((1,2,3),(2,3,4)));
- Calculate a composition series for S4: DisplayCompositionSeries(SymmetricGroup(4));
- DF 4.3 #5, #8, #33
- DF 4.4 #6, #18
- DF 4.5 #3, #40
Recall that the deadline for completing a bonus problem in Friday March 3. (It is not necessary to write up the solution. It is enough to notify the instructor by Friday. It is not necessary to have a complete solution by that date.)
The exam will cover up to and including Section 4.2.
The bonus problems (not to be presented during the exam) are as follows:
- Given the category of finite abelian groups, can you determine which object is which group (up to isomorphism)?
- Determine the group generated by two adjacent faces of the Rubik’s cube.
- DF 3.2 #8, #11,
- DF 3.3 #4, #7,
- DF 3.4 #5, #7, #11, #12
- DF Section 2.4 #9 #17 (Zorn’s lemma appears in Appendix 1), #18
- DF Section 3.1 #1, #12.
- Let F be any field. Show that there is a 1-1 homormophism from to $GL(n,F)$ given by the action of on the indices i in the standard basis . 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 .
- Show algebraically that every isometry T of is a glide reflection provided it has the form $\latex T=T_\lambda A$, where $A\in O(2,R)$, $A\not\in SO(2,R)$, $\latex \lambda\ne 0$.
- Solve the following problems without submitting a solution
- DF Section 2.4 #15, #19
The midterm exam will be held on Wednesday, February 22, 2017.
Bill Rau it the TA for the course. He is available to answer questions about homework in the MAC 10-12 Mondays and 4-5 Mondays.
In class on January 25, we went over a large number of exercises. You will not be asked to submit solutions on February 1. Instead, read all of the exercises from Sections 2.1 (17 problems), 2.2 (14 problems), and 2.3 (26 problems).
Submit a sheet of paper stating for each problem which category each problem falls into:
- (Solved) You have solved the problem.
- (Understand) You have not solved the problem, but you are convinced that you understand exactly how to solve the problem.
- (Unsolved) You are not convinced you understand how to solve the problem.