Math 81
Winter 2006
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 20 - 24, 2006
(Due Wednesday, March 1)
Assignments Made on:
Monday:
  • Study: 14.3, 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
  • 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: nothing assigned


Week of February 13 - 17, 2006
(Due Wednesday, February 22)
Assignments Made on:
Monday:
  • Study: 14.2
  • Do: p567: 6, 7, and the following:
    Suppose that K/F is a finite Galois extension of degree n with Galois group G = {sigma_1, ..., sigma_n}. For an element a in K, define its trace, Tr_{K/F}(a), to be Tr_{K/F}(a) = sigma_1(a) + ... + sigma_n(a). Show that Tr_{K/F} is a surjective mapping from K to F. Hint: first show that there is an element a in K for which Tr_{K/F}(a) is not zero. Note that in characteristic 0 or characteristic p with p not dividing n, this is very easy, but there is a general way to do this in all cases.
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
Friday:
  • Study: 14.3
  • Do: nothing assigned


Week of February 6 - 10, 2006
(Due Wednesday, February 15)
Assignments Made on:
Monday:
  • Study: 14.1
  • Do: Make midterm solutions easy to read and understand.
Wednesday:
  • Study: 14.2
  • Do:
Friday:
  • Study: 14.2
  • Do: nothing assigned


Week of January 30 - February 3, 2006
(Due Wednesday, February 8)
Assignments Made on:
Monday:
  • Study: 13.5
  • Do:
Wednesday:
  • Study: 14.1
  • Do: Take Home exam due Feb 8
Friday:
  • Study: 14.1
  • Do: work on exam


Week of January 23 - 27, 2006
(Due Wednesday, February 1)
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
    Also note that this problem is much easier if you take logs of both sides. Can you justify taking logs?
Wednesday:
  • Study: 13.4, 13.6
  • Do: p 551: 2, 3, 4
Friday:
  • Study: 13.5
  • Do: none


Week of January 16 - 20, 2006
(Due Wednesday, January 25)
Assignments Made on:
Wednesday:
Thursday:
  • Study: 13.2
  • Do: p 531: 19, 20, 21: (for 20, assume the field F is Q, the rationals)
Friday:
  • Study: 13.3
  • Do: none


Week of January 9 - 13, 2006
(Due Wednesday, January 18)
Assignments Made on:
Monday:
  • Study: 13.1, 13.2
  • Do: Finish HW1; start HW2?
Wednesday:
  • Study: 13.2
  • Do: pp 529 - 530: 1, 2*, 3, 4, 5, 14, 16
    *On the last part of 2, do only for a field with four elements.
Friday:
  • Study: 13.2
  • Do: none


Week of January 2 - 6, 2006
(Due Wednesday, January 11)
Assignments Made on:
Monday: (no class)
  • Study:
  • Do:
Wednesday:
  • Study: Chapters 7, 8, 9.1-9.5

    • Note: If you haven't read the homework policy and guidelines, do it now
    Also, this assignment will probably take significant time, so don't put it off!

  • Do: The handout
    • (click here)
Friday:
  • Study: 9.1-9.5, 13.1
  • Do: none assigned


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