TITLE: Motivation behind the proof that $X^4+Y^4+Z^4+W^4$ is unirational
QUESTION [6 upvotes]: I'm trying to understand the  proof that $V(x^4+y^4+w^4+z^4)$ in $\mathbb P^3_k$ is unirational for $k=\overline{\mathbb F_3}$. 
The complete details are in the link so I just write a fast summary, one makes the change $a = w + x$, $b = w − x$, $c = y + z$, $d = y − z$. Then $d=1$, $a=a$, $b=u\cdot a$ and $c=v\cdot a$. And finally one observes that the function field lies inside $k(t,s)$ with $t=a^{1/3}$ and $s=u(a^{4/3}-v)$. 
The algebra involved is straightforward and I understand why we should expect this surface to be unirational. My question is how does one get to those changes of variables? Is it just by trying under the hope that Tate's Conjecture is true?

REPLY [6 votes]: The construction in that bachelor's thesis is taken directly from
the 1974 paper by Shioda
cited as [Sh] in the bibliography (Math. Ann. 211, 233-236),
where it is shown more generally that for any prime power $q$ 
the "Hermitian Fermat surface" $X^{q+1} + Y^{q+1} + Z^{q+1} + W^{q+1} = 0$ 
is unirational over the algebraic closure of the field of $q$ elements.
That's a short paper, and does not indicate how that inseparable
cover was obtained, though it does conclude with the assertion that
such maps can be found also for Hermitian Fermat hypersurfaces
$\sum_{i=0}^{n+1} X_i^{q+1} = 0$ of any dimension $n \geq 2$.
Shioda is still around, so you might be able to learn from him how
he found that transformation.