Name:
Problem Set 9
Math 416, Section 500, Spring 2014
Due: Thursday, April 17th.
Review Sections 53 and 54 in your textbook.
Complete the following items, staple this page to the front of your work, and turn your assignment
in at the beginning of class on Thursday, April 17th. Remember to fully justify all your answers,
and provide complete details.
Definition.
Let F be a field. If f (x) ∈ F[x] is such that every irreducible factor of f (x) is separable over F, then
the splitting field K of f (x) over F is a finite normal extension of F. The Galois group G(K/F) is
called the group of the polynomial f (x) over F.
1. Compute the group of x4 − 1 ∈ Q[x] over Q. This includes checking that every irreducible
factor of x4 − 1 is separable over Q.
2. Compute the group of x3 − 1 ∈ Q[x] over Q. This includes checking that every irreducible
factor of x3 − 1 is separable over Q.
3. A finite normal extension K of a field F is abelian over F if G(K/F) is an abelian group. Show
that if K is abelian over F, and E is a finite normal extension of F such that F ≤ E ≤ K then
K is abelian over E and E is abelian over F.
4. (20 points) Let ω ∈ C be a primitive fifth root of unity.
a. Show that Q(ω) is the splitting field of x5 − 1 over Q.
b. Show that every automorphism of K = Q(ω) maps ω to ωr for some positive integer r.
c. Describe the elements of G(K/Q).
d. Give the group and field diagrams for K over Q as is done in Examples 54.3 and 54.7.
5. Let F be a field, and let n be a positive integer. If char(F) = p > 0, asume that p does not
divide n. Let ζ ∈ F be a primitive nth root of unity. Show that F(ζ) is a finite normal extension
of F, and also prove that G(F(ζ)/F) is abelian.
6. Extra Credit. Use Theorem 54.2 (you do not need to prove this) to show that every finite
group is isomorphic to a Galois group G(K/F) for some finite normal extension K of some
field F.
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Through the course of this assignment, I have followed the Aggie
Code of Honor. An Aggie does not lie, cheat or steal or tolerate
those who do.
Signed:
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