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