Question

1. Find the quotient and the remainder upon dividing 4 + 9i by 3 + i in Z[i].
2.& Determine whether the polynomial& x[sup]4[/sup] - x[sup]3[/sup] - x + 2 is irreducible in Z[sub]3[/sub][x]. If not, factor the polynomial completely
3.& Let F be a field and let p(x) ∈ F[x]. Show that p(x) is irreducible if and only if < p(x)> is maximal.
4.& Consider the ring R = and let p ∈ R. Prove or give a counter example that < p> is prime implies is maximal?

Answer 1

Unfortunately, your question requires a lot of work and cannot be done for free. Submit it with all requirements as an assignment to our control panel and we'll assist you.