1) Find allc ∈ Z3 such that Z3 [x] /(x3 + x2 + c)is a field.
2) Find the degree of and a basis for E = Q
(√2, √3 2, √4 2)
over Q.
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