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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 in $\mathbb{Q}_p$ when $p\equiv3$ p\equiv1$(mod 4) then$\mathbb{Q}(i)$does embed inside in a field with trivial automorphism group. Indeed this is the case for all number fields (finite extensions of$\mathbb{Q}$). Now for an example of an algebraically closed field,$K$, an extension$L$of$K$and an automorphism$\tau$of$K$not extending to one of$L$. Let$K$be the algebraic closure of$\mathbb{Q}$, considered as a subfield of$\mathbb{C}$and let$\tau$be complex conjugation. Let$L=K(x,\sqrt{x^3+ax+b})$be the function field of an elliptic curve$E$over$K$. Each automorphism of$L$takes$K$to itself. Suppose the$j$-invariant of$E$is$i$(considered as an element of$K$). Then any automorphism of$L$taking$K$to itself must fix$i$, and so cannot restrict to$\tau$on$K$. 2 added a further examples There are examples involving the$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}$). Now for an example of an algebraically closed field,$K$, an extension$L$of$K$and an automorphism$\tau$of$K$not extending to one of$L$. Let$K$be the algebraic closure of$\mathbb{Q}$, considered as a subfield of$\mathbb{C}$and let$\tau$be complex conjugation. Let$L=K(x,\sqrt{x^3+ax+b})$be the function field of an elliptic curve$E$over$K$. Each automorphism of$L$takes$K$to itself. Suppose the$j$-invariant of$E$is$i$(considered as an element of$K$). Then any automorphism of$L$taking$K$to itself must fix$i$, and so cannot restrict to$\tau$on$K$. 1 There are examples involving the$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}\$).