is a field if and only if is irreducible if and only if has no zeros in .1) Find all

c ∈ Z3 such that Z3 [x] /(x3 + x2 + c)is a field.

Let then since is an irreducible polynomial which has as a root it means is a basis. This means are linearly independent. Thus, are linearly independent and so they spam a vector space of dimension three.2) Find the degree of and a basis for E = Q