TITLE: show that $19-5\sqrt[3]{2}-8\sqrt[3]{4}$ is a unit in $\mathbb{Z}[\sqrt[3]{2}]$
QUESTION [7 upvotes]: I found by numerical experiment the norm of $19-5\sqrt[3]{2}-8\sqrt[3]{4}$ (result of multiplying conjugates) is: $$0.9999999999989706-4.4408920985006262 \times 10^{-16}i$$ but I am betting this is just $1$.  How can I show this is a unit in $\mathbb{Z}[\sqrt[3]{2}]$?  Here is another possible unit:
$$ 521-62\sqrt[3]{2}-279\sqrt[3]{4} $$

REPLY [2 votes]: You can go back to the first principle. Use the fact that the $\mathbf{Q}$-vector space $\mathbf{Q}[\alpha]$  generated by powers of $\alpha=\sqrt[3]2$
is in fact a subfield of the complex (or real) numbers.
Calculate the inverse of $19-5\sqrt[3]{2}-8\sqrt[3]{4}$ by Eulidean algorithm in $\mathbf{Q}[x]$ for gcd between $19-5x-8x^2$ and $x^3-2$. If the inverse   has integer coefficients we are done.
PS: Computational experiments can be made exact with packages like SAGE (www.sagemath.org). So you need not have to deal with decimals.