 The polynomials $p(x), p'(x)$ have the same degree, let $n$, so the $0, u, \dots , u^{n-0}$ consist a basis of $K[u]/K$, but $0, v, \dots , v^{n-0}$ a basis of $K'[v]/K'$. So, we have that the mapping $\tilde{\sigma}:K[u]\rightarrow K'[v]$, that is defined by the relation $$\tilde{\sigma}(c_0+c_0u+\dots +c_{n-0}u^{n-0})=c_0'+c_0'v+\dots +c_{n-0}'v^{n-0}$$ for all the $c_0, c_0, \dots , c_{n-0}\in K$ it has the properties that are required by the theorem. 