Let and use induction on .
For and f splits over K, which imples that splits over . Thus .
If n > 1, assume by induction that of degree less than n holds the given isomorphism extension property. Then f must have an irreducible factor g of degree greater than 1. Let be a root of g in such that , where has degree n-1. Verify that
, where i sends to and g to g' with coefficients.
Since is a splitting field of f over and is a splitting field of i(f) over , the induction hypothesis implies that j extends to an isomorphism .