Abstract algebra, Field extension

Suppose F and H are fields of size $\;q=p^{r}\;$containing$\;GF(p)\;$as subfield.$\;\alpha\;$is a primitive element of F and $\;\beta\;$ is a primitive element of H.$\;m(x)\;$is the minimal polynomial of $\;\alpha\;.\;$Non-zero Elements of both fields satisfy the equation$\;x^{q-1}=1\; .\;\;\;m(x)\;$is divisor of $\;x^{q-1}-1.\;$Hence there is an element in H (say)$\;\beta^{t}\;\;$which is a root $\;\;m(x)\;.\;$
I want to show that there exists a field isomorphism $\;\phi:F\to H \;$ which carries zero to zero and $\;\alpha\;$to $\;\beta^{t}\;$
Can you help to prove it? I have tried $\;\phi(\alpha^{j} )=\beta^{tj}\;$ but I could not show that $\;\phi\;$ preserves addition.( This argument is presented by Vera Pless in the book Introduction to Theory of Error correcting codes.)