Math 81
Winter 2004

Galois Theory and related topics
Last updated May 31 2008 12:24:47

Homework solutions (Accessible only from on campus)

Week of February 23 - 27, 2004
(Due Wednesday, March 3)
 Monday: Study: 14.4 Do: p 582: 11 Hint: The second part of the question requires a proof or counterexample. To help with the first part, do the following exercise: Consider the symmetric group S_4. It has a normal subgroup A_4 of index 2. Show that A_4 is the only subgroup of S_4 with index 2. Hint: Suppose K is another such subgroup; note that it too is normal. Show that S_4 = K A_4 and the the intersection of A_4 and K has index 2 in A_4 (second isomorphism theorem). Figure 8 from Chapter 3.5 my also prove quite useful. p 589: 1, 3, 8 Note in problem 8, the authors' hint offers one approach to the problem; there are others. Wednesday: Study: 14.4, 14.5 Do:p 595: 1 (and determine its degree), 2, and the following generalization of (2): Let m_1, ..., m_r be pairwise coprime square free integers with m_i > 1. Find a primitive element which generates the extension Q(sqrt(m_1), ... , sqrt(m_r))/Q, and of course prove that it is a primitive element. Friday: Study: 14.5 Do:none

Week of February 16 - 20, 2004
(Due Wednesday, February 25)
 Monday: Study: 14.2 Do: p567: 6, 7 Wednesday: Study: 14.2 Do: pp 581 - 582: 3, 4 (assume the ground field is Q) 7 (with x^4 -2 instead of x^8 - 2 (full Galois correspondence to be done in class Wednesday)) 8 (you may want to read a bit about "p-groups" in your text) 9 (too easy, drill sergeant) Friday: Study: 14.3 Do:none

Week of February 9 - 13, 2004
(Due Wednesday, February 18)
 Monday: Study: 14.2 Do: finish exam Wednesday: Study: 14.2 Do: none Thursday: Study: 14.2 Do: none

Week of February 2 - 6, 2004
(Due Wednesday, February 11)
 Monday: Study: 13.5 Do: nothing new Wednesday: Study: 14.1 Do: Takehome exam goes out; due next Wednesday Friday: Study: 14.1 Do: work on exam

Week of January 26 - 30, 2004
(Due Wednesday, February 4)
 Monday: Study: 13.4, 13.6 Do: pp 555-556: 1, 3, 5, 7, 10 Note in #7, there is an obvious typo d|n should be d|m Wednesday: Study: 13.4, 13.6 Do: p 551: 2, 3, 4 Friday: Study: 13.5 Do: none

Week of January 19 - 23, 2004
(Due Wednesday, January 28)
 Wednesday: Study: 13.2 Do: The handout (click here) Thursday: Study: 13.3 Do: none Friday: Study: 13.4 Do: p545: 2, 3, 4

Week of January 12 - 16, 2004
(Due Wednesday, January 21)