How to construct a given group as multiplicative group of field

Question

Question:

Assume an abelian group $G$ is given. Then, is there a field $F$ such that $F^{\times} \cong G$ ? What is the necessary and sufficient condition about G?

I already know :

1. If $G$ is finite and $\# G + 1$ is not power of any prime, then such a F can not exist.

2. The multiplicative group of finite field is cyclic. So, if finite group $G$ is not cyclic, $F$ can not exist.