Math 81
Winter 2008

Galois Theory and related topics
Last updated June 24 2010 10:49:57

Announcements:

• Final exam to be distributed on March 7 and collected on March 12

Week of March 3 - 7, 2008
(Due Wednesday, March 12)
 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)
 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)

Week of February 11 - 15, 2008
(Due Wednesday, February 20)
 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)
 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. 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. 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)
 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)
 Wednesday: Study: 13.2 Do: The handout (click here) 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)