In a finite field, is there ever a homomorphism from the additive group to the multiplicative group?
This is impossible for finite fields.
Consider a finite field of order $q$; then the additive group also has order $q$. However, the multiplicative group has order $q - 1$ which does not share any common factors with $q$. Since the order of the image of an element $x$ under a homomorphism must divide the order of $x$ by Lagrange's theorem, it follows that any such homomorphism must be trivial.