I encountered in the book: "Finite Geometry and Combinatorial Applications" the definition of trace and norm functions of an automorphism on a field $\mathbb{F}$ as follows:
Let $\sigma$ be an automorphism on $\mathbb{F}$ of the order $r$ (i.e. $\sigma^r = id$, where $id $ id the identity map).
The trace function of $\sigma$ is defined as $$ Tr_{\sigma}(x) = x + \sigma(x) + \cdots + \sigma^{r-1}(x). $$ Also, the norm function is defined as $$ Norm_{\sigma}(x) = x\sigma(x)\cdots\sigma^{r-1}(x) $$ for every $x \in \mathbb{F}$.
I am wondering if these functions are defined on automorphisms on associate algebras over a field in the same way.
Thanks.