Could someone help with a proof: $F^*$ $I_n$ is normal in $GL_n(F)$.
Notice GL maps to F*:
$det:GL_n(F) \mapsto F^*$ by the determinant map.
And notice it maps back:
$F^* \mapsto GL_n(F)$ by ($a \mapsto aI_n$) where $a \epsilon F^*$
Also note:
Not dot product (so $F^*I_n$ is not a scalar vector, it is a diagonal matrix)