Isomorphic general affine groups implies isomorphic fields

Statement
Suppose $$K_1,K_2$$ are fields and $$n$$ is a natural number. Suppose the general affine groups $$GA(n,K_1)$$ and $$GA(n,K_2)$$ are isomorphic groups. Then, the fields $$K_1$$ and $$K_2$$ are isomorphic fields.

Related facts

 * Isomorphic general linear groups implies isomorphic fields
 * Isomorphic special linear groups implies isomorphic fields

Proof for the case $$n > 1$$
In this case, the additive group of the vector space that's the base of the semidirect product is a minimal normal subgroup, hence is characteristic, and the quotient is isomorphic to the general linear group of degree $$n$$. We can then use the action of the center of this quotient group on this additive group by looking at the orbit of any element of that additive group under the center. This gives us a microcosm of a general affine group of degree one and we can use the proof for the case $$n = 1$$.