Cube map on 3-adic principal units ($(1+3{\mathbb{Z}}_3{)}^3=1+9{\mathbb{Z}}_3$)

The cube of $1+3t$ is $1+9(t+3t^2+3t^3)$. Conversely the polynomial $T+3T^2+3T^3-a$ has a unique simple residue root modulo three for each $a\in {\mathbb{Z}}_3$, so Hensel lemma produces a cube root of every $1+9a$. Thus the principal units modulo their cubes have order three. The change of variable removes the derivative obstruction that prevents direct simple-root lifting for $X^3-u$.