Coefficients in expansion of $(\sqrt[3]{2} - 1)^m$
This is a classical problem first solved by Nagell (Solution complète de quelques équations cubiques à deux indéterminées, J. Math. Pures Appl. 4 (1925), 209–270); for an English version see LeVesque's Topics in Number Theory, vol. II, and I have given a brief account in German here (see the appendix).
The question whether elements in a pure cubic field can have squares in which a coefficient with respect to the basis $\{1, \sqrt[3]{m}, \sqrt[3]{m}^2\}$ is $0$ is related to elliptic curves; see this article.