The ring of Eisenstein integers E = Z[w], w = exp(2πi/3) form a
nice commutative ring. It's "nice" because it's a discrete subring
of the complex numbers C with additive group a rank-2 lattice in C.
(When you identify any two points z, w of C whenever z - w in E,
the quotient of C becomes a torus T with a very specific geometry.)
Namely, T = the result of starting from a regular hexagon and
then sewing together corresponding points on the three pairs of
opposite edges.
Anyhow, the ideals of the ring E are all principal, so of the
extremely simple form J = a E, for some a in E. (The condition
a^2 ≠ a ensures that the ideal is non-trivial.)
Now consider the *quotient ring*
S = E / a E.
If a = K + L w, then the number of points in S is
|S| = |a|^2 = K*K - K*L + L*L.
These {K*K - K*L + L*L} are the same numbers represented by
K*K + K*L + L*L when K, L range over all integers.
Assume gcd(K,L) = 1. Then: What is the quotient ring E / a E ???
The additive portion is just Z / (K*K + K*L + L*L) Z,
nothing surprising.
Question:
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What is the multiplicative structure?
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E.g., for K + L w = 3 + 4w, |S| = 37. What is the best way to describe
the multiplication table? I know how to figure it out, but am looking for
a general formula for the multiplication table of E / (K + Lw) E.
NOTE: The elements of the quotient ring S = E / a E are naturally identified
with the hexagonal tiling of the torus T defined as T = C / a E by the
Voronoi regions of the points of S in T, which is kind of cool.
—Dan
