MATH:5010 - Spring 2017

Homework Assignment 10

Reading (from Dummit and Foote):

Read Sections 13.6, 14.1, 14.2.

Please look through the sections for EACH lecture before the lecture. See the Lecture Schedule for the sections that will be covered.

Then after EACH lecture, review your lecture notes and thoroughly read the sections covered in class.

SCHEDULE CHANGE in the week of April 17:

Monday (04/17): Erik Hasse teaches a discussion session during the lecture in 105 MLH.
Tuesday (04/18): Prof. Bleher teaches a lecture during the discussion session in 118 MH. Part A Problems are due as usual.
Wednesday (04/19): Lecture as usual.
Friday (04/21): Part B Problems are due as usual. THERE WILL BE TWO LECTURES ON FRIDAY, APRIL 21:
The usual lecture on April 21 at 11:30-12:20 in 105 MLH and an EXTRA lecture on April 21 at 5:30-6:20PM in 218 MLH.
(The extra lecture is to make up for Friday, April 28, when I am gone for a conference.)

Part A Problems (to be handed in on Tuesday, April 18, at the beginning of the lecture that takes place instead of the discussion session - see above):

Section 14.1: # 10.
Section 14.2: # 14, 15.
HINT for # 14: Find the minimal polynomial of sqrt(2 + sqrt(2)), show that K = Q(sqrt(2 + sqrt(2))) is a splitting field of this polynomial over Q, and then determine Gal(K/Q).
HINT for # 15: For part (b), one way of doing this is to use the fundamental theorem of Galois theory to see there are precisely 3 intermediate fields of K/F which have degree 2 over F. Then argue with extensions of degree 2, using that char(F) is not 2.

Part B Problems (to be handed in on Friday, April 21, at the beginning of the lecture):

If you hand in ALL Part B Problems together with the Part A Problems already on Tuesday, April 18, 2 extra points will be added to your homework score.

Section 14.2: # 4, 5, 12, 16.
IMPORTANT for #4, #5: Both problems are about the Galois group of f(x) = x^p - 2 OVER Q. In other words, if K is the splitting field of f(x) over Q, then the Galois group of f(x) over Q is equal to Gal(K/Q).
- To find the degree [ K : Q ], Corollary 22 on p. 529 may be useful.
- In #4, "Determine the elements of the Galois group" means that for every element ρ ∈ Gal(K/Q), you need to write down the permutation on the ROOTS of f(x) which ρ defines. If you are able to find generators of Gal(K/Q), it is enough to write down the permutations corresponding to these generators. (Note that Gal(K/Q) can be generated by an element of order p and an element of order p-1. To find the element of order p-1, recall that the unit group of F_p is cyclic of order p-1, say F_p^* = < r > for some integer 1 ≤ r ≤ p-1, and then use this r to define a generator of order p-1.)
- You do not need to find the isomorphism type of Gal(K/Q) in #4. This is done in #5. You are welcome to combine #4 and #5 if you want to.
HINT for #12: Write the roots of f(x) = x⁴ - 14 x² + 9 in the form b, -b, c, -c. Then show that Q(b) is a splitting field of f(x) over Q (look at bc). Note that you probably have to show at some point that f(x) is irreducible over Q. You will have to check for quadratic factors to show this.
IMPORTANT for #16: The field F in parts (c) and (d) is not Q but the field Q(sqrt(3)).

Additional Practice Problems (you have to know how to do these, but you do NOT have to hand these in):

Section 14.1: # 1, 2, 3, 4, 5, 6, 7.

Frauke Bleher
Apr 11 2017