From time to time I ask about units in Cubic fields.  I [noticed](http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/tracenorm.pdf) for $\mathbb{Z}[\sqrt[3]{2}]$ I get an analogue of the Pell equation:

$$ \det \left[ \begin{array}{ccc} a & 2c & 2b \\ b & a & 2c \\ c & b & a \end{array} \right] = a^3 + 2b^3 + 4c^3 - 6abc = 1 $$

without citing the Dirichlet unit theorem.

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Clearly the answers are $a + b\sqrt[3]{2} + c\sqrt[3]{4} = (1 + \sqrt[3]{2} + \sqrt[3]{4})^n$ with $n \in \mathbb{Z}$ (since we this number is unit we can have negative exponent.

Except, I have no way of ruling out other families of solutions.  And is there any constructive way of solving this, i.e. without taking a wild guess?

In the $\text{deg}=2$ case (such as $x^2 - 2y^2 = 1$) there is an answer using the pell eq and the continued fraction $\sqrt{2} = [1;\overline{2}]$.  Can we do something analogous here?

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There are continued fractions you can do on triples of numbers.  I think the first step here is:

$$ (1, \sqrt[3]{2}, \sqrt[3]{4}) \to (1, \sqrt[3]{2}-1, \sqrt[3]{4}-1) \to \dots $$

not sure what the smallest number is here.  I wonder if the familiar story from Pell works here?  Does this euclidean algorithm repeat?  Maybe this does not lead to finding units.

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Since I am basically asking for a <del>walk-through</del> explication of the Dirichlet unit theorem, there are numerous resources that discuss this in generality.

https://mathoverflow.net/questions/19021/avoiding-minkowskis-theorem-in-algebraic-number-theory

[What is your favorite use of the pigeonhole principle?](https://math.stackexchange.com/questions/62565/what-is-your-favorite-application-of-the-pigeonhole-principle)

Another one that comes to mind is Hasse's **Lectures on Number Theory** if you have an English copy of the book.  

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The equation $z^3 - 2 = (z - \sqrt[3]{2})(z - \omega\sqrt[3]{2})(z - \omega\sqrt[3]{2}) = 0$ has a real root and a complex conjugate pair of roots.  So the field $\mathbb{Q}(\sqrt[3]{2})$ has a real embedding and a complex embedding. 

Then define an embedding $K=\mathbb{Q}(\sqrt[3]{2}) \to \mathbb{R}^3$

$$ (a+b\sqrt[3]{2}+c\sqrt[3]{4})(a+ b\omega\sqrt[3]{2}+c\omega^2\sqrt[3]{4})
(a+b\omega^2\sqrt[3]{2}+c\omega\sqrt[3]{4})
= a^3 + 2b^3 + 4c^3 - 6abc$$
 
This is not a familiar norm since it's cubic.  And then define a region:

\begin{eqnarray}
|a+b\sqrt[3]{2}+c\sqrt[3]{4}| &=& c_1 \\
(a+ b\omega\sqrt[3]{2}+c\omega^2\sqrt[3]{4})
(a+b\omega^2\sqrt[3]{2}+c\omega\sqrt[3]{4}) &\leq & c_2^2
\end{eqnarray}

The second equation defines the interior an conic section.

$$ a^2 + \sqrt[3]{2}b^2 + 2\sqrt[3]{4}c^2 + ab + ac + 4bc \leq c_2^2  $$

I am following arguments [here](http://wstein.org/129/lectures/day10/day10.pdf) and [here](http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/unittheorem.pdf) trying to get the case $K= \mathbb{Q}(\sqrt[3]{2})$.  

If $c_1 c_2^2 < A$ can we show there is a lattice point in $\mathcal{O}_K$ inside this 3D region?  This should work in a similar way for $\mathbb{Q}(\sqrt[3]{m})$ maybe the constant $A$ changes.