Hello I was a little bit confused by something and would appreciate any help.

Let $\displaystyle \mathbb{F}$ be a field, and let $\displaystyle \mathbb{K}:\mathbb{F}$ be a finite, normal field extension. So it is the splitting field of some polynomial $\displaystyle f\in\mathbb{F}[x]$.

Let $\displaystyle \theta:K \rightarrow \overline{\mathbb{F}}$ be an $\displaystyle \mathbb{F}$-homomorphism (i.e. it fixes the ground field) into the algebraic closure of $\displaystyle \mathbb{F}$.

I am trying to show that $\displaystyle \theta(\mathbb{K}) = \mathbb{K}$.

So $\displaystyle \mathbb{K}=\mathbb{F}(\alpha_1,\ldots,\alpha_s)$ where $\displaystyle \alpha_i$ are the roots of the polynomial $\displaystyle f$. It follows then that $\displaystyle \theta$ permutes these root since it is an injective field homomorphism, but why does this then automatically mean $\displaystyle \theta (\mathbb{K}) = \mathbb{K}$ ? Is it true that $\displaystyle \mathbb{F}(\theta(\alpha_1),\ldots,\theta(\alpha_s )\)=\theta(\mathbb{F}(\alpha_1,\ldots,\alpha_s))?$ Thanks for any help