## Algebraic Structures, Spring 2021 ### Homework:

Assignment 1, due 1/29:
Section 4, problems 8, 10, 19, 29, 32, 37, 41; Section 5, problems 13, 20, 41, 53, 54

As solve the following: Explain why Z/n is not a group with respect to multiplication mod n. Further, prove that the subset of Z/n consisting of elements that are relatively prime to n is a group under this operation (you might want to recall Bezout’s lemma from Math 381)

Assignment 2, due 2/5:
Section 5, problems 50, 57; Section 6, problems 15, 16, 23, 44, 45, 46, 48, 51, 53, 54, 55

Also, our results from class show that every nontrivial subgroup of the integers Z takes the form kZ for some positive integer k (this is an immediate consequence of our proof that subgroups of cyclic groups are cyclic). Find this positive integer k for the subgroup in problem 45 (in terms of r and s from that problem).

Assignment 3, due 2/12:
Section 8, problems 40, 41, 42, 43 (here S_A is what we called Perm(A) in lecture), 47; Section 9, problems 27, 29, 34, 35

Also, solve the additional six problems posted on Piazza

Assignment 4, due 2/19:
Section 10, problems 4, 28, 33, 37, 39

Also, solve the additional two problems posted on Piazza.

Assignment 5, due 2/26:
Section 13, problems 13, 14, 25, 26, 29, 49

Also, solve the additional two problems posted on Piazza.

Assignment 6, due 3/5:
Section 8, problem 52; Section 13, problems 44, 45, 47, 51; Section 14, problems 27, 30, 31, 34, 40 (for 27 you should read Definition 14.15; Theorem 14.13 will then help with 34)

Assignment 7, due 3/19:
Section 15, problems 35, 36, 37 (here, Z(G) is the center of G; a reminder on the definition is given in this Section), 38, 40

Also, prove the 2nd and 3rd isomorphism theorems by following the steps detailed on Piazza.

Assignment 8, due 3/26:
Section 11, problems 16, 18, 19, 24, 26, 46, 52; Section 15, problems 4, 8, 10, 12; Section 18, problems 7, 12, 52, 53

Assignment 9, due 4/2:
Section 18, problems 44, 46, 55, 56; Section 19, problem 29; Section 21, problems 1, 6, 11, 15

Assignment 10, due 4/9:
Section 26, problems 1, 17, 18, 20, 22, 27; Section 27, problems 2, 24, 28, 34.

Also solve the problem posted to Piazza.

Some remarks: Problem 26.17 requires you to say something about subrings, so here’s a definition: let R be a ring and S be a subset of R, then we say that S is a subring if S is a ring (in it’s own right) *and* the inclusion function S –> R is a ring homomorphism (the latter basically says that the ring structure on S is “induced” from the corresponding one on R). In 26.20, you can assume the ring is a field, since we’ve only defined characteristic in that context.

Assignment 11, due 4/19:
Section 22, problems 24, 29, 30; Section 23, problems 2, 4, 8, 10

Also, solve the problems posted on Piazza (you might want to do the first one before trying problem 24).

Assignment 12, due 4/28:

Section 23, problems 12, 14, 16, 20, 34, 35, 37; Section 27, read Theorem 27.25 (and its proof), then solve problems 18, 19, 30 (we’ll discuss the related Theorem 27.24 on Tuesday)

Assignment 13, due 5/5 (LDOC):

Section 29, problems 2, 4, 30, 32; Section 30, problem 26

Also, solve the two problems posted on Piazza.

Assignment 14 (not collected, but good practice on field extensions):

Section 29, problems 31, 33, 34, 36; Section 31, problems 22, 23, 25, 26, 30