BSM, Spring 2019, Introduction to Abstract Algebra AL1

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Course syllabus

General info

Problem sheets: 1. ; 2. ; 3. ; 4. ; 5. ; 6.

Short summaries: Week 1: Operation, ring ;

Weeks 2-3: Subring, ideal, factor ring, homomorphism ;

Week 4: Direct sum, Number theory in rings ;

Week 5: Gaussian integers, Two Squares Theorem


Homeworks are assigned on Wednesday and are due on next Wednesday. Each problem is one point worth (partial credit is possible), the weekly assignment contains 3 problems from the problem sheets.

HW1. Due on February 20: (i)-(ii) Any 8 of the altogether 28 subquestions from 5(b2)-(h2), except 5e, so that at least four should occur among the labels of the problems out of the six letters b, c, d, f, g, h; (iii) One of the following 5 problems: 3; 4; 5e and 7; 8; 10.

HW2. Due on February 27: (i) Any 2 of the following 4 problems: 1b,c; 6a,c; 18; 20; (ii) Any 6 subquestions of 12; (iii) 13(a2)-(a3) or 14a,c or 15.

HW3. Due on March 6: (i) Any 2 subquestions of 22b,c,e,f, and any 2 subquestions of 23a-d; (ii) Any 2 pairs of 24a-d or 29a-d; (iii) Any 4 of the altogether 9 subquestions in 27 and 28.

HW4. Due on March 13: (i) Any 4 of the following altogether 11 questions: 31b, 31c, 32a, 32b, 32c, 32d, 33b, 33c, 33d, 33e, 33f; (ii)-(iii) Any 2 of the following 6 questions: 34abc, 34d, 35b, 35c, 36a, 38a (here T is not required).

HW5. Due on March 20: (i) Any 4 parts of 47; (ii) One part of 46; (iii) One of 41, 43, and T in 38a.

HW6. Due on March 27: Any 3 of the followng 6 problems: 48b,c; 49b,c; 52; 53; 54 (without part (*)); 55.

Extra problems


You can hand in the solutions of these problems without time limit.

X1. Prove that if an associative operation has a left identity e and every element c has a left inverse c' with respect to e, then e is also a right identity and c' is also the right inverse of c.

What happened in class?

February 11: We investigated some basic properties of binary operations and their relation to each other. We defined rings and fields and saw various examples. We solved Problems 1a, 2a,b,c,d, and started to discuss 5a.

February 13: We discussed Problems 5a, 5(b1), and 6b.

February 18: We discussed equivalence relation, and applied congruences to prove a-b|a^n-b^n and to solve Problem 19. We solved also Problem 9. During the office hour we took several nice rings into our hands.

February 20: We discussed previous HW problems 3, 4a, 5(b3), 5(d5), 7, and 10. For subrings and ideals, we solved 11, 13(a1), and 16a.

February 25: After a few remarks concerning the HW just returned, we introduced ring isomorphism, and noted that 5(c5) is isomorphic to the field of the real numbers and 5(c9) is isomorphic to the complex field. Then, introducing the finitely generated ideals, we solved Problems 16c, 17b and c. During the office (half an) hour, the squirrels (Problem 20) proved to be the main attraction.

February 27: We dealt with factor rings and ring homomorphisms: we solved Problems 21, 22a,d, 26, and 30.

March 4: After proving the homomorphism theorem (Problem 30a), we turned to direct sums of rings: we solved 31a and 33a. Preparing the number-theoretical investigations in general rings, we summarized the basic notions concerning divisibility for the integers and for various types of polynomials: unit, irreducible, division algorithm, gcd, prime, and unique prime factorization. During the office hour, factor rings were the stars.

March 6: We discussed Problems 22e,f, 23, and 24b1 in detail. Concerning number theory in general rings, we introduced the basic notions, discussed 34e, 35a, and stated the theorems about UFT, PID, and Euclidean rings (36b, 37b, and 38b). Proofs will come later.

March 11: We examined the relation between various notions and facts about divisibility: we saw how the gcd of two elements is related to the ideal generated by them (Problem 37a), how the gcd guarantees the uniqueness of prime factorization (part of Problem 36b), and learned why the ideals were originally introduced by Kummer to attack more efficiently Fermat's Last Theorem. As a preparation to handle which integers are the sum of two squares, we determined which integers are the difference of two squares (part of Problem 39). The office hour was "ideal" in directing us directly to the direct sums.

March 13: We proved 38b, and then developped the basic number theory among the Gaussian integers (Problems 40, 42, and 45).

March 18: We proved some interesting properties of the ring D of rationals with odd denominators: there is just one irreducible element apart from assoiates, we discussed why Euclid's proof about the infinitude of primes fails in D, UFT is true in D, moreover D is a Euclidean ring (parts of Problems 35c, 36a, and 38a). We solved 44 (which had been used for showing that the primes of the form 4k+1 are not Gaussain primes). As an application of Gaussian integers, we proved the Two Squares Theorem. Gaussian integers were the topic also during the office hour (for those who did not go to have some rest after the exhausting travels during the long weekend).

March 20: We studied some basic facts about finite fields via solving Problems 48a, 49a, 50, and 51 (except for proving 51b).