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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...".

Martin Sleziak
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user2554
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  • If you meant ${x,y,z \in \mathbb{Z}, F(x,y,z) = \pm 1}$ then it is $\mathbb{Z}[n^{1/3}]^\times$ not $O_{\mathbb{Q}(n^{1/3})}^\times$, for example $O_{\mathbb{Q}(10^{1/3})} = \mathbb{Z}+ 10^{1/3}\mathbb{Z} + (\frac13+\frac23 10^{1/3}-\frac13 10^{2/3})\mathbb{Z}$ so $O_{\mathbb{Q}(10^{1/3})}^\times \cong {x,y,z \in \mathbb{Z}, F(x+\frac13 z,y+\frac23 z,-\frac13 z) = \pm 1}$ – reuns Feb 09 '19 at 18:54
  • @reuns - I didn't understand your last comment. What are you actually asking? – user2554 Feb 10 '19 at 09:22
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    The question is now posted also on MO: How did Gauss find the units of the cubic field $Q[n^{1/3}]$?. I think that this answer gives a very reasonable advice about [meta-tag:cross-posting]. Probably the most important point is that each copy should be linked to the other one. – Martin Sleziak Feb 11 '19 at 13:52

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All we know is entry 137 in his diary, where he announces on Dec. 23, 1808, that he has begun studying the theory of cubic forms as well as the solution of the equation $$x^3 + ny^3 + n^2z^3 - 3nxyz = 1; $$ in addition there are fragments in his papers containing the table with solutions, which are printed in his Collected Works. Since Gauss did not publish anything on cubic forms, this is not related to his other work, except that it is a generalization of his theory of binary quadratic forms in Section V of his Disquisitiones.

Of course the solvability of the equation above as well as the fact that each solution arises (up to sign) by taking powers of a fundamental solution is a special case of Dirichlet's unit theorem.

  • thanks for the answer! i voted it although it didn't help me a lot. I'm aware that Gauss never published anything on cubic forms, so i tried to understand what was his procedure from his unpublished fragments. If it's not possible to infer from his nachlass what was it, than i'd like to get a little bit explanation of the modern way to solve this equation. I guess you already know the fragment "Zur Theorie der complexen Zahlen", but i'll add the link to it here just for help - http://gdz.sub.uni-goettingen.de/pdfcache/PPN23599524X/PPN23599524X___LOG_0056.pdf. – user2554 Feb 11 '19 at 15:40
  • For modest values of $n$ one computes solutions of the equation $F(x, y, z) = m$ for small values of $m$ and then forms qotients of the corresponding numbers in the field hoping that one of the quotients is integral and produces a unit. Computing units in general algebraic number fields is a challenging problem; see the books by Henri Cohen. In Gauss's case I would guess he worked with diophantine approximation, but my speculation is as good as anyone's. I can't see how to do it with cubic forms. –  Feb 12 '19 at 14:31
  • Solving $F(x,y,z)=1$ isn't too difficult. See: https://math.stackexchange.com/questions/3481203/units-in-cube-root-system/3483357#3483357 – O. S. Dawg Sep 21 '20 at 00:53