Math 366–001 HW 7, Spring 2015
This assignment is due Monday, April 27 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. (1 pt) Show that x2 + x + 4 is irreducible over Z11 .
2. (2 pts) Find all zeros and their multiplicities of x5 + 4x4 + 4x3 − x2 − 4x + 1 over Z5 .
3. (2 pts) Let V = R3 and W = {(a, b, c) ∈ V | a2 + b2 = c2 }. Is W a subspace of V ? If
so, what is its dimension? If not, why not?
4. (2 pts) Let V = R3 and W = {(a, b, c) ∈ V | a + b = c}. Is W a subspace of V ? If so,
what is its dimension? If not, why not?
5. (1 pt) 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.)
6. (1 pt) Same setup as #1: Write E in the form a0 + a1 + . . . + ak k | a0 , . . . ak ∈ Q ,
for some number and some integer k.
7. (1 pt) Same setup as #1: Write E in the form
.
Q[x]
I
for some ideal I of Q[x].
1