Remark 2

In the notation of the preceding Corollary there exist in general $K$-automorphisms of $\Omega$ distinct from the identity. Hence there is in general no uniqueness about the isomorphism $v$ of $\Omega$ onto $\Omega'$ extending the isomorphisms $u$ of $K$ onto $K'$. For similar reasons there is in general more than one isomorphism of a splitting extension $E$ onto a splitting extension $E'$ for the same family $(f_i)_{i \in I}$ of polynomials. We recall that by contrast, for the perfect closure we have uniqueness (V, p. 5).