Abstract Algebra II

Instructions. Please solve the following problems (show all your work). You can use your textbooks,
class notes. Work on your own and do not discuss the problems with your classmates or anyone else.
Please let me know if you have any questions concerning the problems or you do need some hints.
1. (10 pts) Show that Z2 [x] =
x2 + x + 1 is a Öeld. List all of its elements and Önd multiplication
and addition tables.
2. (10 pts) Show that the given number  2 C is algebraic over Q by Önding f(x) 2 Q[x] such that
f() = 0.
a)  = 1 + p2
b)  = pp 3 2 i
3. (10 pts) Classify the given number  2 C as algebraic or transcendental over the given Öeld F . If
is algebraic, Önd deg(; F).
a) Let  = i and F = Q:
b) Let  = p and F = Q.
4. (10 pts) Classify the given number  2 C as algebraic or transcendental over the given Öeld F . If
is algebraic, Önd deg(; F).
a) Let  = p and F = Q().
b) Let  = 2 and F = Q().
5. (10 pts) Solve the following problems.
a) Find the degree and a basis for the given Öeld extension Q p2; p3; p18
over Q: Justify your
answers.
b) Find the degree and a basis for the given Öeld extension Q p2; p 3 2
over Q: Justify your answers.
c) Prove that Q p2 + p7
= Q p2; p7
:
1
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