Math 366–002 HW 7, Spring 2014
This assignment is due Monday, April 28 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. If A and B are ideals of some ring, prove that AB ⊂ A ∩ B.
2. If I is an ideal of a ring R and I contains a unit, show that I = R.
3. Find the four zeros of x2 + 3x + 2 in Z6 .
4. List all polynomials of degree 2 in Z3 [x].
5. Determine the quotient and remainder when dividing x3 + 2x + 4 by 3x + 2 in Z5 [x].
6. Construct an isomorphism ψ : R[x] → S[x] out of an isomorphism φ : R → S. In
particular, define the map ψ and show that ψ(p(x) + q(x)) = ψ(p(x)) + ψ(q(x)) and
ψ(p(x)q(x)) = ψ(p(x))ψ(q(x)).
7. Show that x2 + x + 4 is irreducible over Z11 .
8. Find all zeros and their multiplicities of x5 + 4x4 + 4x3 − x2 − 4x + 1 over Z5 .
9. 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?
10. 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?
1