Question about group $Aut(F/K)$

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I have an open question: if $F$ is an extension field of $K$, and $|Aut(F/K)|=[F:K]=p$ with $p$ is a prime and $p\neq char(K)$. So $F$ is a Galois extension of $K$ or not (if the answer is no, please give me an example)