I am familiar with the partial trace operator that one deals with in Quantum computing on density matrices. That would be sum of diagonal elements. I was reading a section on finite fields and polynomials that defined a trace operation from prime subfield ($\mathbb{F}_{p^t}$)to the base field ($\mathbb{F}_{p}$) as follows:
$$ \text{tr}[a] = \sum_{l=0}^{t-1} a^{p^l} $$
I am unable to relate this to the intuition I am familiar with, can someone provide me with some explanation how I can connect this with the linear algebraic definition of trace
Generally, if $L$ is a finite field extension of a field $K$ of degree $n$, meaning that $K \subset L$ and the dimension of $L$ as a vector space over $K$ is $n$, then we can associate to any $a \in L$ the left multiplication operator
$$M_a : b \mapsto ab$$
which is a $K$-linear map $L \to L$ and hence which defines an element of $\text{End}_K(L) \cong M_n(K)$ (where this identification comes from choosing some basis of $L$ over $K$). Then we can define a $K$-valued trace for elements of $L$ by defining
$$\text{tr}_{L/K}(a) = \text{tr}(M_a).$$
This trace can be computed in any basis and does not depend on the choice of basis. It is equal to the sum of the roots of the characteristic polynomial of $M_a$.
If furthermore $L$ is a finite Galois extension of $K$ with Galois group $G$, then the roots of the characteristic polynomial of $M_a$ turn out to be $ga$ as $g$ runs across all elements of $G$, from which it follows that
$$\text{tr}_{L/K}(a) = \sum_{g \in G} ga.$$
Finally, $\mathbb{F}_{p^n}$ is a finite Galois extension of $\mathbb{F}_p$ of degree $n$ with Galois group the cyclic group of order $n$ generated by the Frobenius map $x \mapsto x^p$, from which it follows that
$$\text{tr}_{\mathbb{F}_{p^n}/\mathbb{F}_p}(a) = \sum_{i=0}^{n-1} a^{p^i}$$
as desired.