Trace of a product in finite fields

Question

Let $\mathbb{F}_{p^n}$ be a finite field of cardinal $p^n$. We choose an arbitary generator $t$ of $\mathbb{F}_{p^n}$ such that $\mathbb{F}_{p^n}=\mathbb{F}_p(t)$, then any element $a$ in $\mathbb{F}_{p^n}$ can be written as $a_0+a_1t+...+a_{n-1}t^{n-1}$.

For any two elements $a,b$ in $\mathbb{F}_{p^n}$, Then I want to ask if we can prove $\operatorname{tr}(ab)=\sum_0^{n-1}a_ib_i$.

Thanks.