## Homework Assignments

(main course page)

Homework 1, due Friday September 5: 1.7, 1.30, 1.31, 1.32

Homework 2, due Monday September 8: 2.1, 2.4, 2.6

Homework 3, due Wednesday September 10: 2.2, 2.5, 2.10, and prove that if an abelian group G contains elements x, y of orders m, n respectively, and l is the least common multiple of m and n, then G contains an element of order l.

Homework 4, due Friday September 12: 2.11, 2.17, 2.25, and the following: Let m and n be positive integers, and let K and L be the mth and nth cyclotomic fields. Determine the composite field KL and the intersection of K and L.

Homework 5, due Monday September 15: 1.16, 2.19, 2.20

Homework 6, due Wednesday September 17: 2.22, 2.27, 2.28

Homework 7, due Friday September 19: 2.8, 2.23, 2.29

Homework 8, due Monday September 22: 2.35, 2.37, 2.40

Homework 9, due Wednesday September 24: 3.1, 3.4, 3.7

Homework 10, due Friday September 26: (1) Let R be a commutative ring with identity and let I_1, I_2, ..., I_n be pairwise relatively prime ideals in R (that is, I_i + I_j = R for all i≠j). Let J be the intersection of all I_i. Prove that (a) J is the product of all the I_i, and (b) R/J is isomorphic to the product of all the R/I_i. (2) Prove that if R is a Dedekind domain with only finitely many prime ideals, then R is a principal ideal domain.

Homework 11, due Monday September 29: 3.8, 3.9

Homework 12, due Wednesday October 1: 3.10, 3.11

Homework 13, due Friday October 3: 3.12, 3.13

Homework 14, due Monday October 6: 3.14, 3.18

Homework 15, due Wednesday October 8: 2.21, 3.19

Homework 16, due Friday October 10: 2.34

Homework 17, due Monday October 13: 3.20, 3.21

Homework 18, due Wednesday October 15: 3.29, 3.30

There will be no homework due on Friday October 17, Monday October 20, or Wednesday October 22 due to the midterm exam.

Homework 19, due Friday October 24: 4.5, 4.6, 4.7

Homework 20, due Monday October 27: 4.9, 4.10, 4.11

Homework 21, due Wednesday October 29: 4.8, 4.12, 4.15

Homework 22, due Friday October 31: 4.13, 4.14

Homework 23, due Monday November 3: 5.1, 5.6, 5.11

Homework 24, due Wednesday November 5: 5.2, 5.3, 5.5

Homework 25, due Friday November 7: 5.4, 5.8, 5.13

Homework 26, due Monday November 10: 5.31, 5.33, 5.34

Homework 27, due Wednesday November 12: 5.35, 5.36

Homework 28, due Friday November 14: 5.37, 5.38

Homework 29, due Monday November 17: 6.1a, 6.1b

Homework 30, due Wednesday November 19: 6.4

Homework 31, due Friday November 21: 6.5

Homework 32, due Monday November 24: 6.10, 6.11

Homework 33, due Monday December 1: 6.2, 6.3

Homework 34, due Wednesday December 3: 7.3, 7.4, 7.5

Homework 35, due Friday December 5: 7.15

Homework 36, due Monday December 8: 7.16