# ring

1. ### Zero divisors of a polynomial factor ring

Problem As we've observed, a(x) = x^2 + x + 1 2 Z_3[x] is reducible. According to the theory (Theorem 17.5), Z3[x]=(a(x)) is not a field. Find a non-0 element q(x)+(a(x)) that has no inverse. Suggestion: Find a 0-divisor of Z3[x]=(a(x)). Attempt at a solution: I've factored x^2 + x + 1 into...
2. ### Polynomial Ring of a Field

I have had some trouble with the second part of this problem. Any help would be greatly appreciated. My problem is this: If F is a field, then (x) is a maximal ideal in F[x], but it is not the only maximal ideal. Now, I have proven (I think), that F[x]/(x) \cong F and thus (x) is maximal, but...
3. ### How many zero divisors in a ring

If I want to know how many zero divisors there are in the ring \mathbb{Z}_5[x]/(x^3-2), how do I go about solving that? I've noted that for p(x) = x^3-2, p(3) = 0 \space \space (\mathrm{mod} 5) and p(5) = 0 \space \space (\mathrm{mod} 5). Does that make 3 and 5 zero divisors? If I could write...