The proof for (b) will do this one.

Say that where are non-units. It follows that and so thus . Let then . Thus, which means . If then but is the only square mod 3. Thus, rational primes which are of form are Eisenstein primes.b) a rational prime of the form 3k+2, where k is a positive integer, is an Eisenstein prime

Say that has form . By using quadradic reciprocity it follows that . Thus, there is an integer so that thus . Thus, .c) a rational prime of the form 3k+1, where k is a positive integers, factors into the product of two primes that are not associates of one anotherIfwere prime then it would divide both those factors but that is impossible since then , since it means . This is impossible and so is reducible. Thus, . We have and so . Since the norm is a rational prime it means are prime. Note, . And so factors into two prime which are not associate.