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Department of Mathematics
University of Pittsburgh
MATH 1250 (Abstract Algebra)
Midterm, Spring 2014
Instructor: Kiumars Kaveh
Last Name:
Student Number:
First Name:
TIME ALLOWED: 50 MINUTES. TOTAL: 50 + 2 bonus
NO AIDS ALLOWED. WRITE SOLUTIONS ON THE SPA
MATH42122 Galois Theory.
Exercises IX.
1. Let E be the splitting eld inside C of the polynomial f = x5 2, and let
G = (E : Q).
(a) Show that E = Q(, ) where = 5 2 and = exp( 2i ).
5
(b) Show that (E : Q) = 20, and deduce that |G| = 20.
(c) Show that G is
MATH42122 Galois Theory.
Exercises VIII.
A Larger Example of the Galois Correspondence
Let E be the splitting eld of the polynomial f = x4 2 over Q.
1. Show that E = Q(, i) where =
4
2.
2. Prove that (E : Q) = 8.
3. Show that the Galois group G = (E : Q)