Let $F$ be a finite field of order $p^n$. Let $K$ be a degree $d$ extension of $F$. Let $F^+$ and $K^+$ denote the additive groups of $F$ and $K$, respectively. Let $F^*$ and $K^*$ denote the multiplicative groups of units of $F$ and $K,$ respectively. Compute $Ext^i_{\mathbb{Z}}(F^+, K^+)$ and $Ext^i_{\mathbb{Z}}(F^*, K^*)$ for all $i$.
I know that Abelian groups can be viewed as Z-modules and all previous Ext computations I have done involved simple Z-modules so this problem seems very out of the ordinary to me. My guess as to what the projective resolution of $F^+$ is $$\cdots Z \rightarrow Z \rightarrow (Z/pZ)^n \rightarrow 0$$
EDIT:
After applying $Hom(\_,(Z/pZ)^{nd})$, we have $$0 \rightarrow Hom((Z/pZ)^n,(Z/pZ)^{nd}) \xrightarrow{d_0} (Z/pZ)^{nd} \xrightarrow{d_1} \cdots $$
Any hints?