Hello everyone and thanks in advance for all the helpers.
I'm trying to develop an intuition for finite fields and I've stumbled upon the following question (it's not a homework question) which I cannot answer myself (my abstract algebra is very rusty).
Let $q=p^h$ for some prime number $p$ and let $x,y\in\mathbb{F}_q^n$. It is known that $x$ and $y$ can be viewed as $x',y'\in\mathbb{F}_p^{hn}$. Is it true that $\langle x',y' \rangle =0 $ if and only if $\mathrm{tr}(\langle x,y \rangle)=0$?
By $\langle x, y\rangle$ I mean $\sum_{i=1}^nx_iy_i$ (and similarly $\langle x',y'\rangle = \sum_{i=1}^{hn}x'_iy'_i$).