Definition: A french field is a field where the additive group is isomorphic to the multiplicative group.

Question: Can there exist a French Field?

Partial Answer: Since the multiplicative group does not contain zero (additive identity) we have that the multiplicative group is aproper subgroupof the additive group. Thus, if there exists an isomorphism then there exists a bijective map from the additive group to the multiplicative group. Thus, by definition of cardinality the additive group has the same cardinality as the multiplicative group. Since, the multiplicative group is aproper subgroupof the additive group then by the definition of an infinite set we see that the field CANNOT be a finite field.