MathNet 1988 — Part 01 Question 20

Let $k^{3} = 2$ and let $x$, $y$, $z$ be any rational numbers such that $x + y k + z k^{2}$ is non-zero. Show that there are rational numbers $u$, $v$, $w$ such that $(x + y k + z k^{2})(u + v k + w k^{2}) = 1$.