Lifts mod pᵏ

$\mathbb{Z}/p^k$'s multiplication table, its elements listed by their base-$p$ digits, the lowest first: by residue mod $p$, then mod $p^2$, and so on. The table nests: $p \times p$ blocks, each a copy of $\mathbb{Z}/p^{k-1}$'s table, whose cells are the $p^2$ products of the lifts of one pair of residues. Reducing mod $p^{k-1}$ collapses each block back to one cell.

Colored by $p$-adic valuation, the zero divisors gather in the blocks of multiples of $p$, then of $p^2$, a self-similar pattern. Select a cell to light its value's lifts: the $p$ values congruent to it mod $p^{k-1}$, or all those congruent to it mod $p$. On the diagonal, light the square roots: for odd $p$, each square root of a unit mod $p$ lifts to exactly one mod $p^k$ (Hensel's lemma), so a unit square has two roots at every level.

Multiplication in , n = , listed by p-adic digits, each product colored by its p-adic valuation, 1 white. Selecting a cell lights its ; shift-click to select several.

The additive side, and the other ways of listing, are in Addition mod n.