Let the minimal polynomial for $\alpha$ over $K$ be $X^d+c_{d-1}X^{d-1}+…+c_1X+c_0$. Prove that: $$Tr_{K(\alpha)/K}(\alpha)=-c_{d-1}, N_{K(\alpha)/K}(\alpha)=(-1)^dc_0$$
I found this question on an online document while learning trace and norm by myself. It contains a proof for a more general result with a lot of theorems and settings that I'm not familiar with. Is there an easy way to prove just this special case with basic knowledge about Galois extension?
Take the $K$-vector space basis $1, \alpha, \alpha^2, \dotsc, \alpha^{d-1}$ of $K(\alpha)$.
Multiplication by $\alpha$ in $K(\alpha)$ is represented by matrix \begin{equation*} \vec{A} = \begin{pmatrix} 0 & 0 & 0 & \dots & 0 & -c_0 \\ 1 & 0 & 0 & \dots & 0 & -c_1 \\ 0 & 1 & 0 & \dots & 0 & -c_2 \\ 0 & 0 & 1 & \dots & 0 & -c_3 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 1 & -c_{d-1} \end{pmatrix} \end{equation*} a $d \times d$ matrix whose characteristic polynomial is $f_\alpha(t)$.
From the characteristic polynomial, we can read off the trace and norm, giving the required result.