Math 366–002 HW 8, Spring 2014
This assignment is due Wednesday, May 7 in class. Feel free to work together, but be
sure to write up your own solutions.
As for writing it up, please write legibly on your own paper, including as much justification
as seems necessary to get the point across.
1. Let E be the splitting field of x3 − 1 over Q. Write E in the form Q(a), for some
number a. (Google “cyclic roots of unity” in case it isn’t clear to you how to find the
solutions of x3 − 1 = 0.)
2. Same setup as #1: Write E in the form a0 + a1 + . . . + ak k | a0 , . . . ak ∈ Q , for
some number and some integer k.
3. Same setup as #1: Write E in the form
Q[x]
.
I
for some ideal I of Q[x].
4. Write the splitting field of x4 + x2 + 1 = (x2 + x + 1)(x2 − x + 1) over Q in the form
Q(a) for some number a.
5. Let E be a finite extension of R. Use the fact that C is algebraically closed (meaning
that all algebraic extensions of C have degree 1) to prove that E = C or E = R.
√ √ √
6. Assume that Q( 2, 3 2, 4 2, . . .) is an algebraic extension of Q. Show that it is not a
finite extension (so finite implies algebraic, but not vice versa!).
7. Find [GF (1024) : GF (2)]. (Corollary 1 of Chapter 22 could help!)
8. Find [GF (1024) : GF (32)].
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