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There are examples involving the $p$-adics: for instance $\mathbb{Q}_p$ itself has trivial automorphism group. Indeed as $\mathbb{Q}(i)$ embeds inside $\mathbb{Q}_p$ when $p\equiv3$ (mod 4) then $\mathbb{Q}(i)$ does embed inside a field with trivial automorphism group. Indeed this is the case for all number fields (finite extensions of $\mathbb{Q}$).