Recently I read jstor article "Gauss and the Early Development of Algebraic Numbers", which gives a good description of the genesis of Gauss's ideas regarding the foundations of algebraic number theory. Among other pieces of useful information, it mentions a certain ternary cubic form which Gauss studied in 1808 in connection with his attempts to understand the underlying principles of higher reciprocity laws (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 arises as the norm of the number $x+vy+v^2z$ (where $v = n^{1/3}$) in the pure cubic field created by adjoining $v$ the the field of rationals. Since Gauss wanted to know where this expression equals 1, this investigation can be interpreted as an attempt to find the units (numbers of norm 1) in this cubic field. Gauss than recorded the units for certain values of n, and in some instances exhibited the fundamental unit.

I didn't find enough information about this investigation of Gauss. So now, to my questions:

  • What was Gauss's procedure? and how does it relate to Gauss's other investigations in algebraic number theory?

  • Does it relate in some aspects to Dirichlet's unit theorem? I ask because this article says that Gauss's investigation was "a step in the progression from Lagrange to Dirichlet, the latter of whom in 1842-46 developed the general theory of algebraic units...".


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