Show that if the field $k$ is generated over $f$ by the elements $\alpha_1,\dots,\alpha_n$ then an automorphism $\sigma$ of $k$ fixing $f$ is uniquely determined by $\sigma(\alpha_1),\dots,\sigma(\alp ha_n)$. in particular, show that an automorphism fixes $k$ if and only if it fixes a set of generators for $k$