## finite fields

Find two distinct monic irreducible polynomials f(x) and g(x) of degree 3 over the finite field F_3.(for instance, let f(x)=x^3 + 2x + 1 and g(x)= x^3 + 2x + 2.) Let a be a root of f(x) and be be a root of g(x).

i)identify a root of f(x) in F_3(b) (this means we adjoin b to F_3) and similarly identify a root of g(x) in F_3(a). (I didn't understand what this part of the question means.) Then write an explicit isomorphism from F_3(a) to F_3(b).

ii) identify all the three roots of f(x) in F_3(a) and that of g(x) in F_3(b). (Okay, i can do this by trail and error, but is there a better procedure for these type of questions?)

Thanks!