Cube map on 3-adic principal units
= Cube map on 3-adic principal units
{title2=$(1+3\mathbb Z_3)^3=1+9\mathbb Z_3$}
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 $X^3-u$.