Recently I read the National Mathematics Magazine article "Bell - 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.
I checked in Gauss's work and the relevant pages are p.21-26 from volume 8, where he denotes the norm of an algebraic (cubic) integer $t$ by $\varphi(t)$. In 1 he established several basic properties of the norm function (under "Theorem I"); that the norm function is homogenous and multiplicative. In [2] he writes down the minimal polynomial of $t = x+vy+v^2z$ (when only integer coefficients are allowed) in his notation, which in our notation translates into:
$$t^3-3xt^2+3(x^2-nyz)t-(x^3 + ny^3 + n^2z^3 - 3nxyz)=0,$$
note that the norm $\varphi(t)$ is the free coefficient, so it equals $t\cdot t_1\cdot t_2$ (and this is exactly the definition of the algebraic norm). In [3] he characterizes prime numbers that are representable by this cubic form as those for which $n$ is a cubic residue.
At the end of this fragment there are two numeric tables, the first gives the fundamental unit of $\mathbb Q[2^{1/3}]$, together with its inverse unit and their first nine powers, and the second table gives the units reciprocal to the fundamental units of the cubic field $\mathbb Q[n^{1/3}]$ for certain values of $n$ between $2$ and $37$. As franz lemmermeyer remarked in an answer to the same question posted on math.stackexchange, the solvability (in integers) of the equation above as well as the fact that each solution arises by taking powers of the fundamental solution (the fundamental unit) is a special case of Dirichlet's unit theorem.
While verifing the correctness of some data in the second numeric table (using some tables available on web), I noticed that the calculations involved relatively large numbers, so I guess searching for these reciprocal fundamental units without some useful algorithm is very computationally inefficient. Therefore, I believe Gauss might have used some algorithm to increase searching efficiency. I added the second table here, just to illustrate the complexity invloved.
What is most interesting to me is a remark by Fricke on a certain strange handnote Gauss wrote next to [2]:
In Gauss's handwriting you can find the following note
- Correction of $v$ is roughly: $$-\frac{D^2}{9BC^2}$$
Here $D$ is Gauss's notation for the norm (he replaces $\varphi$ with $D$), and $A= b^2-ac,B=nc^2-ab, C=a^2-nbc$ (where $t = a+bv+cv^2$). $A,B,C$ are actually related to $t$ through the relation: $t^{-1} = C+Bv+Av^2$. This note is "strange" to me because I don't understand what is meant by "correction of $v$" — isn't $v$ supposed to be constant ($v=n^{1/3}$)?
Despite not having an idea what is the meaning of correction, one thing I noticed is that it is an homogeneous function of the form. That is, the norm $D$ is homogeneous of degree $3$, $B$ and $C$ are homogeneous of degree $2$, and therefore $-\frac{D^2}{9BC^2}$ is homogeneous of degree $0$. This means that the correction is invariant with respect to multiplication of the cubic integer $t$ with a real integer; that is, if $(x,y,z) = x+vy+v^2z$, than it's supposed "correction" is the same as for every integer multiple of $(x',y',z') = (\frac{x}{gcd(x,y,z)},\frac{y}{gcd(x,y,z)},\frac{z}{gcd(x,y,z)})$. I hope this invariance property might help give a clue about the kind of arithmetic proccess for which Gauss needed to make corrections in the value of $v$.
So my questions are:
- What was Gauss's procedure? and how did he find fundamental units?
- What is the meaning of Gauss's note (about the correction of $v$)?
