isomorphism function

Question

Find an isomorphism $f \colon Q(\zeta^2 _5 )\to Q(\zeta^3 _5)$ that is not the identity function.

I do not know how to solve it when $f$ is not defined like $f(x)=y.$

I know to show $f$ is an isomorphism I first need to show it is a homomorphism which means $f(a+b)=f(a)+f(b)$ and $f(ab)=f(a)f(b)$, then I need to show $f$ is both one-to-one and onto and I know $\zeta _5 = e^{(2\pi i)/5}$.

Could you please help me how can I solve it?

Answer 1

Hint: Prove that complex conjugation is an automorphism on $\mathbb{C}$. Then prove that it maps $ζ_5^2$ to $ζ_5^3$.