Let $K \subset L$ be a field extension of a finite field. Let $\alpha \in L$ be a non-zero root of the polynomial $f = \sum_{i=0}^{m} a_{i}X^{i} \in K[X]$. Prove $Tr_{L/K}(\alpha^{-1}) = -a_{1}a_{0}^{-1}$.
Where $Tr_{L/K}(x) = \sum_{\sigma \in Aut_{K}(L)} \sigma(x)$ for $x \in L$, is the field trace. I have already proven the identity $Tr_{L/K}(\alpha) = -a_{m-1}a_{m}^{-1}$, but I'm having difficulty proving this property of the trace. Previously, I had written $f$ as a product of linear factors and rewrote the product to obtain an expression for the coefficients. But I'm unsure whether this is the right approach when proving this identity. I also know that $Tr_{L/K}\colon K \rightarrow L$ is a surjective homomorphism, hence $Tr_{L/K}(\alpha^{-1}) = Tr_{L/K}(\alpha)^{-1}$ but I'm unsure how to proceed.
Any help is greatly appreciated. Thanks in advance.