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.

---

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?

---

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.

---

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

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

[What is your favorite use of the pigeonhole principle?](http://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.  Otherwise I found [Artin-Whaples](https://projecteuclid.org/euclid.bams/1183507128), and I have to work out the real and complex embedding of $K = \mathbb{Q}(\sqrt[3]{2})$ - and I have trouble deciding if their Axiom 2 and Axiom 2a are related to more modern terminology.

I could work through as much and show where I got stuck?