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

Week of February 24 - 28, 2003
(Due Wednesday, March 5)
 Monday: Study: 14.3, 14.4 Do: p 569: 1, 3, 8 Note: In 8, ignore the comment in brackets unless you can figure out how to use it. Wednesday: Study: 14.4 Do: p576: 1, 2, and the generalization of (2): Let m_1, ..., m_r be pairwise coprime squarefree 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: TBA

Week of February 17 - 21, 2003
(Due Wednesday, February 26)
 Monday: Study: 14.2 Do:p562: 3, 4 (you should assume the ground field is Q in these two problems) Wednesday: Study: 14.2, 14.3 Do: p562: 7 (with x^4 - 2 instead of x^8 - 2; note we will do the full Galois correspondence for this polynomial on Monday or Wednesday), 8 (you may want to look up a bit on "p-groups"), 9 (this should be easy), 11 The second part requires a proof or counterexample. To do 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 of index 2. Hint: Suppose K is another such subgroup. Note that it too is normal. Show that S_4 = K A_4 and that the intersection of A_4 and K has index 2 in A_4 (subhint: second isomorphism theorem). The diagram on page 112 may also prove useful. Friday: Study: 14.3 Do: TBA

Week of February 10 - 14, 2003
(Due Wednesday, February 19)
 Monday: Study: 14.2 Do: Takehome exam distributed; due 10am Friday. Wednesday: Study: 14.2 Do: none Friday: Study: 14.2 Do: none

## Homework Assigments

Week of February 3 - 7, 2003
(Due Wednesday, February 12)
 Monday: Study: 13.5, 14.1 Do: none Wednesday: Study: 14.1 Do: none Thursday: Study: 14.1 Do: none (plan ahead; take home exam next week)

Week of January 27 - 31, 2003
(Due Wednesday, February 5)
 Monday: Study: 13.4 Do: pp535 - 536: 1, 3, 5, 7, 10 Wednesday: Study: 13.5 Do: pp531 - 532: 2, 3, 4 Friday: Study: 13.5, 13.6 Do: none

Week of January 20 - 24, 2003
No class Monday, Martin Luther King Day
(Due Wednesday, January 29)
 Wednesday: Study: 13.2, 13.3 Do: From last week's homework, you should know the degree of the extension . Determine a basis over for this extension. Rather than try to directly prove the set you write down is linearly independent, give an argument, (based on what we have done in class), which proves your set is a basis. Your argument should extend easily to , for which you should simply state (with confidence) a basis. Determine the degree of the extension , and write down three intermediate fields (fields between and the top field). Thursday (xhour): Study: 13.3, 13.4 Do: none Friday: Study: 13.4 Do: p 525: 1, 2, 3, 4

Week of January 13 - 17, 2003
(Due Wednesday, January 22)