Math 81
Winter 2008
Galois Theory and related topics
Last updated June 24 2010 10:49:57

General Information Syllabus HW Assignments


Announcements:



Week of March 3 - 7, 2008
(Due Wednesday, March 12)
Assignments Made on:
Monday:
  • Study: 14.7

  • Do:
Wednesday:
  • Study: 14.7

  • Do:
Friday:
  • Study: 14.6

  • Do: Final Exam


Week of February 25 - 29, 2008
(Due Wednesday, March 5)
Assignments Made on:
Monday:
  • Study: 14.3

  • Do: p 582:
    7 (with x^4 -2 instead of x^8 - 2 (Galois correspondence was done in class))

    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,
    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: 14.7

  • Do: nothing additional


Week of February 18 - 22, 2008
(Due Wednesday, February 27)
Assignments Made on:
Monday:
Wednesday:
  • Study: 14.2

  • Do: nothing additional
Friday:
  • Study: 14.2

  • Do: nothing additional


Week of February 11 - 15, 2008
(Due Wednesday, February 20)
Assignments Made on:
Monday:
  • Study: 14.1

  • Do: Finish homework due Wednesday;
    Start Midterm exam
Wednesday:
  • Study: 14.1

  • Do: Midterm due Friday
Friday:
  • Study: 14.1, 14.2

  • Do: nothing new


Week of February 4 - 8, 2008
(Due Wednesday, February 13)
Assignments Made on:
Monday:
  • Study: 13.5

  • Do: The following two problems
    Note: This short homework assignment is due Feb 13; your midterm is due the 15th. Plan ahead.
    1. Let K be a field of prime characteristic p, and suppose that L/K is a finite extension with [L : K] = m where p does not divide m. Show that L/K is separable.
    2. Let K be a field of odd prime characteristic p, a an element of K and suppose that x^p - a has no roots in K. Show that x^{p^n} - a is irreducible over K for all n >= 1. Hint: You definitely want to think inductively and consider carefully the case of n=1. Bigger hint: If adjoining a root of a polynomial of degree d produces an extension of degree d, what can be said about the polynomial?
Wednesday:
  • Study: 13.5

  • Do: nothing new
Thursday:
  • Study: 14.1

  • Do: nothing new


Week of January 28 - February 1, 2008
(Due Wednesday, February 6)
Assignments Made on:
Monday:
  • Study: 13.4, 13.6

  • Do: p. 545: 3, 4
Wednesday:
  • Study: 13.4, 13.6

  • Do: p. 555-6: 1, 3, 5, 10
Friday:
  • Study: 13.5

  • Do: p. 551: 2, 3, 4


Week of January 21 - 25, 2008
(Due Wednesday, January 30)
Assignments Made on:
Wednesday:
Thursday:
  • Study: 13.3

  • Do:no more assigned
Friday:
  • Study: 13.3, 13.4

  • Do: no more assigned


Week of January 14 - 18, 2008
(Due Wednesday, January 23)
Assignments Made on:
Monday:
  • Study: 13.2

  • Do: Put finishing touches on hw1
Wednesday:
  • Study: 13.2

  • Do: pp 529-530: 1, 2*, 3, 4, 5, 14, 16
    * for the last part of 2, do only for a field with 4 elements.
Friday:
  • Study: 13.2

  • Do: no more assigned


Week of January 7 - 11, 2008
(Due Wednesday, January 16)
Assignments Made on:
Monday:
  • 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!

    Here is a short handout most of which we will review in class, but very useful for the homework.

  • Do: The handout
    • (click here)
Wednesday:
  • Study: 9.1-9.5, 13.1

  • Do: no more assigned
Friday:
  • Study: 13.1, 13.2

  • Do: no more assigned


Thomas R. Shemanske
Last updated June 24 2010 10:49:57