BSM, Fall 2021

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Course syllabus

General information

Extra office hour in zoom: Every Sunday 7PM.

Attention, please! There will be an office hour on October 18 (Monday) 7PM on zoom instead of the office hour on October 17 (Sunday).

Midterm is on October 21 12:05-14:00. The October 19 class will be a review session discussing the problems in the sample midterm .

There will be an extra office on October 20, 7 PM on zoom.

Homework info: There will be 11 homework sets, each containing three parts, and each part is 1 point worth. Partial credit is possible. So you can achieve at most 33 HW points and a maximum of 30 counts into your final grade.

Homework is assigned on Thursday and is due on next Thursday (written or typed) at the beginning of the class. Please, justify your answers in detail. Besides the oral hints given in class you can get futher help in the second hour of Tuesday classes, in the zoom office hour on Fridays 7PM, and via email.

September 7: handout 1 and problem sheet 1 . Divisibility, congruences, phi(n). We solved problems 1 (3 proofs), 3a (2 proofs), 4, 5b, 5d (2 proofs), and 11a. We determined the possible remainders of a cube when divided by 7.

September 9: handout 2 and problem sheet 2 . Euler--Fermat theorem, linear congruence, linear Diophantine equation, binomial coefficient, binomial theorem. We solved problems 11b (2 solutions), 12a, 14a,b, 17a,b, 18a, and 18d (2 proofs). We revisited the possible remainders of a cube when divided by 7.

HW1 (due 09/16 before class): (i) Any four of the eight parts in 2 and any four of the eight parts in 6; (ii) Any two of the next seven problems: 3/(ii)-(iii); 5a,c; 7; 8; 9; 10a; and 10b; (iii) Any two of the five problems 18b; 18c; 19a; 19b; and 19c.

September 14: In the first half of the class, we had a thorough discussion through many examples about algebraic operation, associative, commutative, and distributive laws, identity, and inverse. We saw that one of the most important operations, composition, is associative but not commutative. We solved Exercises 20 and 21a,b,c. In the second half of the class we had an intensive problem solving session concerning the number theory hw problems.

September 16: handout 3 and problem sheet 3 . Rings and fields, many examples. Zero divisors. We solved 24a,h1, and 25a,b.

HW2 (due 09/23 before class, handed in or sent by email in pdf format): (i) Any two of the five problems 12b; 13b; 14c; 15a; and the first question of 16; (ii) Seven questions of 25 (except (a) and (h1) done in class), at least four different letters should be involved and at least five should be rings, indeed; (iii) Any two of the six problems 20bc; 22; 23; 25c; 26; and 29. Xtra problems: 15b and the second question of 16.

September 21: Subrings. We solved 6gh, 7, 10, 18b, and 30a,b1,c. The second half of the class was dedicated to problems mostly about rings.

September 23: problem sheet 4 . Ideals and factor rings. We solved 34(a1)(a2), 35a, 38, 39a,b, 40, and 41a.

HW3 (due 09/30 before class, handed in or sent by email in pdf format): (i)-(ii) Any eight of the eleven problems in 31; (iii) Any three of the six problems 32a; 32b; 36a; 36b; 37a; and 37b.

September 28: Factor ring: the operations on the cosets are well defined, i.e. they are independent of the selection of the representatives. Also, we discussed HW2 problems 23, 24(b1),(c1), and (c5). The second half of the class was ideal for problem solving about ideals and subrings.

September 30: Factor rings and ring homomorphisms. We solved 42e, 45a,b, and 47a.

HW4 (due 10/07 before class, handed in or sent by email in pdf format): (i) Any four of the twelve problems 41b-f, 42a-d,f-h; (ii) Any two pairs of the four pairs in 43 (or 48 equivalent to 43 by the homomorphism theorem and the natural homomorphism); (iii) Two of the four parts in 46, at least one of them should be a homomorphism, and one of the four parts in 47b.

October 5: handout 4 and problem sheet 5 . We summarized factor rings and ring homomorphisms and their relation. We gave a second solution to 42e using the homomrphism theorem and discussed HW problems 31j, 32b, 36a, and 37a. The problem solving session in the second half of the class provided excellent occasions for factor rings and cosets to appear as protagonists.

October 7: Direct sum of rings. Introduction to number theory in rings: integral domain, divisibility, units, division algorithm, gcd, irreducible elements, unique factorizations theorem, principle ideal domain, examples for comparison: integers (Z), even numbers (2Z), polynomials over a field (F[x]), and polynomials over the integers (Z[x]). We solved 50a,b, 52a, 53a,(c1),(c2),(c3),e and revisited 31b and 43(a2) using direct sums.

HW5 (due 10/14 before class, handed in or sent by email in pdf format): (i) Two parts of 51; (ii) Two parts of 52b,c,d,e,f; (iii) One of 53b,(cd),d, and one of 55a and 50c.

October 12: handout 5 and sample midterm . We summarized the basic facts about number theory in integral domains: divisibility, units, gcd, division algorithm, irreducible and prime elements, unique factorization, principal ideal domain, Euclidean domain. We solved problems 43b, 54a, 56a, and 57b. The problem solving session in the second half of the class was a direct approach to direct sums of rings.

October 14: handout 6 and problem sheet 6 . Gaussian integers. We solved 67, 68, 73, and 74b.

HW6 (due 10/28 before class, handed in or sent by email in pdf format): (i) 74a or c; (ii)-(iii) Four of the altogether nine parts of 57d and 75, plus one of 69, 71a, and 71b.

October 19: sample midterm . We discussed 58-64.

October 21: Midterm.