This is the sketch of the proof.

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

.