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

Let be a field, and let be a finite, normal field extension. So it is the splitting field of some polynomial .

Let be an -homomorphism (i.e. it fixes the ground field) into the algebraic closure of .

I am trying to show that .

So where are the roots of the polynomial . It follows then that permutes these root since it is an injective field homomorphism, but why does this then automatically mean ? Is it true that Thanks for any help