Isomorphic special linear groups implies isomorphic fields

Statement
Suppose $$K_1,K_2$$ are fields and $$n > 1$$ is a natural number. Then, if the special linear groups $$SL(n,K_1)$$ and $$SL(n,K_2)$$ are isomorphic groups, $$K_1$$ and $$K_2$$ must be isomorphic as fields.

Note that the case $$n = 1$$ is different, because $$SL(1,K)$$ is the trivial group for any field $$K$$.

Related facts

 * Isomorphic special linear groups implies isomorphic fields
 * Isomorphic general affine groups implies isomorphic fields
 * Isomorphic unitriangular matrix groups implies isomorphic fields