How many elements in the finite field?

Question

Let $\mathbb F_{q^{2n}}$ be a finite field, and let $a,b\sim\text{Unif}\lbrace 0,q^{2n}-1\rbrace$. What is the probability that:

1. $a - a^q\in\mathbb F_{q^2}$

2. $a - a^q = b - b^q$

Answer 1

The additive version of Hilbert's theorem 90 tells you that the kernel of the trace $\mathbb{F}_{q^{2n}} \rightarrow \mathbb{F}_{q}$ consists precisely of the elements of the form $a - a^q$ with $a \in \mathbb{F}_{q^{2n}}.$ Likewise, the elements of $\mathbb{F}_{q^2}$ of the form $a - a^q$ are also precisely the traceless elements.

Since the trace is a nonzero functional, its kernel has order $q^{2n-1}$ (on $\mathbb{F}_{q^{2n}}$) and $q$ (on $\mathbb{F}_{q^2}$). Combined with the linearity of the map $a \mapsto a - a^q$ this is enough to solve both problems.