## Spring-08. Math H113. Honors Introduction to Abstract Algebra

Instructor: Alexander Givental , TuTh 9:30-11, 85 Evans (CCN: 54727)
• Office hours: Tu 2-4 p.m., in 701 Evans
• Texts - Required: --> I. N. Herstein, Topics in Algebra, 2nd ed., Wiley & Sons
Recommended: R. Solomon, Abstract Algebra , Brooks/Cole.
• Grading: Final (40%)+Quizzes (30%)+Homework (30%)
• HW: Weekly homework assignments are posted to this web-page, and your solutions will be due on Thursdays in class (the first one is due Jan. 31).
• GSI: Arturo Prat-Waldron aprat@math.berkeley.edu will hold office hours (for all sections of Math 113) at 891 Evans on W 9-2, and Th 9-11, 3:30-6:30, starting Th, Jan. 24.
• Final: Monday, May 19, 2008, 8-11 a.m. (Exam Group 10).
HW1 (due Th, Jan 31): 1.1(6,11a), 1.2(7), 1.3(7b,10a,17), Construct a one-to-one correspondence between two sets A and B, given a one-to-one correspondence between A and a subset of B, and a one-to-one correspondence between B and a subset of A.
HW2 (due Th, Feb 7): 2.3(1c, 3, 7, 11, 25), Compute the determinant of the nxn-matrix with all entries on the diagonal equal to 2, right under and right above the diagonal -1, and 0 everywhere else.
HW3 (due Th, Feb 14): 2.5(3,5,15,29), 2.7(2), Let G be a group, G' be another group such that as a set G'=G, and for all a,b in G, the product ab in G' is defined to be the product ba in G. Are the groups G and G' isomorphic?
HW4 (due Th, Feb 21): 2.6(1,2,8,13), 2.7(5ab,19). Beware that all problems except the 1st (and possibly 2nd) one are non-trivial. Here is one more problem; my 9-year old son got it from his math tutor: A number, whose rightmost digit is 2, doubles when the 2 is moved to the leftmost place. Find at leat one such a number.
HW5 (due Th, Feb 28): 2.7(13,14) 2.8(7a,16) 2.10(22)
A. Prove that the group of rotations of the cube is isomorphic to S_4.
B. Prove that the group of all automorphisms of Z_n is isomorphic to the multiplicative group Z_n^* (of invertible remainders modulo n).
HW6 (due Th, March 6): 2.8(5), 2.9(8), 2.10(11,13), 2.11(7)
Show that the group of rotations of a dodecahedron is isomorphic to the alternating group A_5.
HW7 (due Th, March 13): 2.11(17), 2.12(5,8,20), 2.13(5)
Prove that if p^k (p prime) divides the order of a group G, then G has a subgroup of order p^k.
HW8 (due Th, March 20): 2.13(4ab,7,11,15,16) 2.14(6)
HW9 (due Th, April 3): Read 3.1, 3.2. Solve 3.2(3,7,11,12,14)
Solve: N knights at the King Arthur's round table are served unequal amounts of cereal. At discrete moments of time T=1,2,3, ... each knight takes a half of the amount of cereal from each of his neighbor's plates. (Respectively, at this very moment the contents of his own plate goes into the plates of his neighbors.) Describe the distribution of cereal at large moments of time T=2007, 2008, ...
HW10 (due Tue, April 15): 3.4(3,12,20) 3.6(5bdef,6)
HW11 (due by Th, April 17): Suppl. Problems in Ch.3(4,8)
A. Prove that every non-constant polynomial equation f(x,y)=0 with complex coefficients has infinitely many complex solutions (x,y).
B. It can be easily derived from Hilbert's Nullstellensatz that if a system of complex polynomial equations f_1=...=f_k=0 has no complex solutions, then the ideal generated by f_1,...,f_k coincides with the whole polynomial ring C[x_1,...,x_n]. Give a counter-example to the same statement over reals.
C. Describe all maximal ideals of: (a) C[x,y]; (b) R[x]; (c) R[x,y] (R stands for the field of real numbers, C for complex).
HW12 (due by Th, April 24): 3.7(7,8) 3.8(3b,4,6)
A. Represent 1105=5 x 13 x 17 as the sum of two squares in all possible ways.
B. Prove that the product p_1...p_n of distinct primes, all of which have the remainder 3 modulo 4, cannot be represented as the sum of two squares.
HW13 (due by Th, May 1): 3.9(3,7) 3.10(2,5) Suppl. Probl. in Ch. 3(18)
Prove that the multiplicative group of any finite field is cyclic.
HW14 (due by Th, May 8): 5.1(3a,4), 5.4(7c,8,13)
Prove that if for relatively prime m and n, regular m-gon and n-gon are constrictible, then the regular mn-gon is also constructible.
Answers and Hints to Homework Problems
Solutions to the Final Exam Problems