Math 81
Winter 2005

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

Homework solutions (Accessible only from on campus)

Week of February 21 - 25, 2005
(Due Wednesday, March 2)
 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 14 - 18, 2005
(Due Wednesday, February 23)
 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 7 - 11, 2005
(Due Wednesday, February 16)
 Monday: Study: 14.1 Do: Wednesday: Study: 14.2 Do: Friday: Study: 14.2 Do: nothing assigned

Week of January 31 - February 4, 2005
(Due Wednesday, February 9)
 Monday: Study: 13.5 Do: Wednesday: Study: 14.1 Do: Take Home exam due Feb 9 Friday: Study: 14.1 Do: work on exam

Week of January 24 - 28, 2005
(Due Wednesday, February 2)
 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 17 - 21, 2005
(Due Wednesday, January 26)
 Wednesday: Study: 13.2 Do: The handout (click here) 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 10 - 14, 2005
(Due Wednesday, January 19)