Build the idea in three steps
- The stage of the second case is 𝓞 K — the Eisenstein integers, the ring that contains a cube root of unity ζ₃ with 1 + ζ₃ + ζ₃² = 0.
- Inside this world, a sum of two cubes splits cleanly: a ^ 3 + b ^ 3 = (a + b) (a + η b) (a + η² b). The displayed calc verifies the algebra, with the rewrites using η³ = 1 and the vanishing of 1 + η + η².
- A later lemma shows that λ² divides one of these three factors. Further lemmas transform the solution so that the selected factor becomes a + b, while preserving the required hypotheses. This is more than renaming the variables.
A small example
Over the ordinary integers, x² − y² = (x + y)(x − y) is the familiar split. Enriching the ring buys the same gift for a sum of cubes: the cube roots of unity are exactly what makes three factors appear.