Math 81
Winter 2004
Galois Theory and related topics
Last updated May 31 2008 12:24:47

General Information Syllabus HW Assignments


Homework solutions (Accessible only from on campus)

Week of February 23 - 27, 2004
(Due Wednesday, March 3)
Assignments Made on:
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)
Assignments Made on:
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)
Assignments Made on:
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)
Assignments Made on:
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)
Assignments Made on:
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)
Assignments Made on:
Wednesday:
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)
Assignments Made on:
Monday:
  • Study: 13.1, 13.2
  • Do: pp 529-530: 1, 2*, 3, 16
    *On the last part of 2, do only for a field with four elements.
Wednesday:
  • Study: 13.2
  • Do: pp 529 - 530: 4, 5, 14
Friday:
  • Study: 13.2
  • Do: none


Week of January 5 - 9, 2004
(Due Wednesday, January 14)
Assignments Made on:
Monday:
  • Study: Notes from class (Chapter 9)
    Note: If you haven't read the homework policy and guidelines, do it now
    Also, this part of the assignment will probably take significant time, so don't put it off!
  • Do: The handout (click here)
Wednesday:
  • Study: 7.5 and (lightly) 15.4
  • Do: The handout (click here)
Friday:
  • Study: 13.1
  • Do: none assigned


Thomas R. Shemanske
Last updated May 31, 2008 12:24:47 EDT