Any suggestions using the minimal polynomial?
How about $D_{\mathbb{Q}_{p}[\phi]/\mathbb{Q}_{p}}=(-1)N_{\mathbb{Q}_{p}[\phi],\mathbb{Q}_{p}}(f')$? But foremost I prefer you suggest me a correct method. I want to avoid the computational approach.
The approach is to find the change of basis matrix. The basis is $\{1,\phi,...,\phi^{p-1}\}$ and we take a random element $a=a_{1}+a_{2}\phi+...+a_{p-1}\phi^{p-1}$. Then multiply it by each of the basis elements and represent it in the basis form to get the matrix. Then evaluate that determinant.