Galois: Why does $\{1,f,f^2\}$ correspond to $\mathbb{Q}(\omega)$?

Question

I'd like to discuss this example from the wikipedia article.

It claims that the group $\{1,f,f^2\}$ corresponds to $\mathbb{Q}(\omega)$. But then it must hold true that the fixed field $Fix(\{1,f,f^2\})=\mathbb{Q}(\omega)$ by Galois correspondence.

I see that happening for $1,f$ but for $f^2$ it's $f(\omega)^2=\omega^2$, isn't it? It's not fixed.

Where am I thinking wrong?

Answer 1

Well but you are conflating $(f(\omega))^2=f(\omega)×f(\omega)=\omega×\omega$ with $f^2(\omega)=f(f(\omega))=f(\omega)=\omega$.