Suppose $F_1$ and $F_2$ be two finite fields such that additive groups of $F_1$ and $F_2$ are isomorphic and also multiplicative groups of $F_1$ and $F_2$ are isomorphic.Does this imply that $F_1$ and $F_2$ are isomorphic as fields ?(Without using uniqueness of finite fields)
Any hints/ideas ?
Note that the fact that the additive groups are isomorphic immediately implies that the multiplicative groups are isomorphic, since the multiplicative groups are cyclic, and so if the additive groups are isomorphic, then the fields must have the same number of elements, and thus the multiplicative groups have the same number of elements two.
Of course the uniqueness of finite groups solves the problem as soon as it is posed, so I guess you are wondering if it is possible to prove elementarily that any additive homo/iso-morphism between fields is automatically a multiplicative homo/iso-morphism too. That is not the case since the additive isomorphism $x\mapsto -x$ is not a multiplicative isomorphism.