## nt.number theory – How did Gauss find the units of the cubic body \$ Q[n^{1/3}]\$?

I recently read an article on jstor "Gauss and the early development of algebraic numbers", which describes the genesis of Gauss's ideas on the foundations of the algebraic theory of numbers, among other useful information, he mentions a certain ternary cubic form that Gauss studied in 1808 to try to understand the principles under higher rates of reciprocity (cubic reciprocity in this case).

The particular form is:
$$F (x, y, z) = x ^ 3 + ny ^ 3 + n ^ 2z ^ 3 – 3nxyz$$ and Gauss attempted to find (rational) solutions to the Diophantine equation $$F (x, y, z) = 1$$. As the article explains, this particular form appears as the norm of the number $$x + vy + v ^ 2z$$ (or $$v = n ^ {1/3}$$) in the pure cubic field created by joining $$v$$ the field of rationals. Since Gauss wanted to know where this expression was equal to 1, this investigation can be interpreted as an attempt to find the units (norm 1 numbers) in this cubic field. Gauss then recorded the units for some values ​​of n and, in some cases, presented the fundamental unit.