Make Explicit isomorphism of two fields

Question

In picture, How can we find explicit isomorphism?

What i have done is that the two finite fields are GF(16) and isomorphic. But i dont know how to find explicit isomorphism pi(x)

i want to know somewhat general way

Answer 1

To find an explicit isomorphism, you should find a root of $ P(x) = x^4 + x^3 + x^2 + x + 1 $ in $ K = \mathbf F_2[x]/(x^4 + x^3 + 1) $. Working in the quotient, we have

$$ P(x+1) = (x^4 + 1) + (x^3 + x^2 + x + 1) + (x^2 + 1) + (x + 1) + 1 = x^4 + x^3 + 1 = 0 $$

so that $ x + 1 $ is a root of $ P(x) $ in $ K $. Now, the surjective homomorphism $ \mathbf F_2[x] \to K $ given by the evaluation map at $ [x+1] $ descends to an isomorphism $ \mathbf F_2[x]/(x^4 + x^3 + x^2 + x + 1) \to K $ of the quotient, which is the explicit isomorphism you are looking for.