$[F:K]$ finite , $Aut_K(F)=\{f_1,...,f_n\}$ , there is $a\in F$ such that $\{f_1(a),...,f_n(a)\}$ forms a basis for $F$ over $K$ ; is $F/K$ Galois? Let $F/K$ be a finite extension field , let $Aut_K(F)=\{f_1,...,f_n\}$ , suppose there exist $a\in F$ such that $\{f_1(a),...,f_n(a)\}$ forms a basis for  $F$ over $K$ ; then is $F/K$ Galois extension ? 

Yes. The given conditions imply that $ [F : K] \leq |\textrm{Aut}(F/K)| $, and this is a necessary and sufficient condition for a finite extension of fields to be Galois (in fact, this is how being Galois is defined in some sources).
