Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 1 / 1 / a

Interpret the map as the action on ${\mathbb{C}}^n$. A smooth representation requires every vector to have an open stabilizer in the locally profinite group ${\mathrm{GL}}_n(F)$. Take the first standard basis vector $e_1$. Every identity neighbourhood contains matrices $d_a=\mathrm{diag}(1+a,1,\dots ,1)$ with $a\ne 0$ sufficiently small in the local field. We may require $|a|<1$, so $1+a\ne 0$. Injectivity of the field embedding givesThus no identity neighbourhood fixes $e_1$, and its stabilizer is not open. Consequently the representation is not smooth for any $n\ge 1$. This includes $n=1$, when a nonzero vector's stabilizer is just the identity. The field embedding need not be continuous, but no continuity assumption would repair this stabilizer obstruction. The existence of the stated embedding itself forces characteristic zero; in positive characteristic no field embedding into $\mathbb{C}$ exists.