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.

I have not found enough information about this Gaussian investigation. So now, to my questions:

  • What was the Gauss procedure? And how does this relate to Gauss's other investigations in the algebraic theory of numbers?

  • Does this have anything to do with the Dirichlet Unity Theorem?? I ask the question because this article says that the Gaussian investigation was "a step in the progression of Lagrange to Dirichlet, the latter having developed in 1842-1846 the general theory of algebraic units …".

abstract algebra – On the irreducibility of a polynomial lemma and Gauss

Let $ P (X) = X ^ 5 – 6X + 3 $.

Prove that it is irreducible on $ mathbb {Q} $.

In the solution that I have for this exercise, I literally have:

I miss the necessary knowledge about irreducibility by reducing modulo p. All I know is that there is a bijection between the roots of P in $ overline { mathbb {Q}} $ and the roots of $ overline {P} $ in $ overline { mathbb {F} _p} $.

I also know this generalization of Lemma de Gauss: in a factorial ring A, with its fractional body K, a primitive $ P $ is irreducible in $ A[X] Leftrightarrow $ $ P $ is irreducible in K[X].

How does this apply to P above?

Thank you for any direction or help.