A field is rigid if it has no nontrivial automorphisms. A number field is a finite extension of $\mathbb{Q}$. My question is, are there any rigid number fields?
I’m pretty sure that if $K$ is a finite Galois extension of $\mathbb{Q}$, then $K$ cannot be rigid. But what about non-Galois extensions of $\mathbb{Q}$?
Every quadratic number field is Galois so the smallest possible counterexample is cubic. A cubic field is rigid iff it's not Galois: the simplest example is $\mathbb{Q}(\sqrt[3]{2})$.