I am trying to find all the isomorphisms from the representation modulo p(x), where $$p(x) = X^3 + X^2 + X + 2,$$ to the representation modulo q(Y), where $$q(y) = Y^3 + 2Y^2 + 1,$$ in $$ GF(3^3) $$
I believe there will be 3 of them. Below is what I have done so far:
$$ (a + bX + cX^2)(d + eX + kX^2) = (a,b,c)(d,e,k) = ad + (ae + bd)X + (ak+be+cd)X^2 + (bk+ce)X^3 + ckX^4 $$
After reducing the above to p(x): $$(a,b,c)(d,e,k) = (ad-2(bk+ce-ck),ac+bd-ck-bk-ce,ak+be+cd-bk-ce)$$
After reducing the same thing but to q(x) now: $$(a,b,c)(d,e,k) = (ad-(bk+ce-2ck),ae+bd-ck,ak+be+cd-2(bk+ce-2ck))$$
Isomorphism needs to satisfy the following property: $$f(a,b,c) * f(d,e,k) = f((a,b,c)*(d,e,k))$$
I assume I need to define $$f(x) = u + vX + wX^2$$ and express f(a,b,c) in terms of 'u's,'v's and 'w's but this is where I get confused and don't know what to do.