[math-fun] Quotient rings of the ring of Eisenstein integers
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: ----- What is the multiplicative structure? ----- 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
I believe it's isomorphic to the ring Z / (K*K + K*L + L*L) Z. In particular, your constraint of gcd(K, L) = 1 means that n = K*K + K*L + L*L is only divisible by primes of the form 6k + 1, and that there's no pair of conjugate prime factors p, p' of K + omega L. Consequently, we can apply the Chinese Remainder Theorem (over the Eisenstein integers, which are a Euclidean domain!) to reduce this to a product of rings of the form E / a E, where a is a power of a non-real Eisenstein prime. Then it's additively isomorphic to Z / |a|^2 Z, and the existence of a 6th root of unity in the ring Z / |a|^2 Z gives a way to homomorphically identify it with E / a E (where the sixth roots of unity are the obvious ones). Then apply the Chinese Remainder Theorem (over Z) to combine all of those prime-power rings together. -- APG.
Sent: Tuesday, May 21, 2019 at 4:23 AM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun@mailman.xmission.com Subject: [math-fun] Quotient rings of the ring of Eisenstein integers
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: ----- What is the multiplicative structure? -----
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
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Dan, Take a look at *On Sublattices of the Hexagonal Lattice <http://neilsloane.com/doc/Me201.pdf>* M. Bernstein, N. J. A. Sloane and P. E. Wright, *Discrete Math.*, 170 (1997), pp. 29-39. (it's on my home page, see http://neilsloane.com/doc/Me201.pdf) On Tue, May 21, 2019 at 6:20 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
I believe it's isomorphic to the ring Z / (K*K + K*L + L*L) Z.
In particular, your constraint of gcd(K, L) = 1 means that n = K*K + K*L + L*L is only divisible by primes of the form 6k + 1, and that there's no pair of conjugate prime factors p, p' of K + omega L.
Consequently, we can apply the Chinese Remainder Theorem (over the Eisenstein integers, which are a Euclidean domain!) to reduce this to a product of rings of the form E / a E, where a is a power of a non-real Eisenstein prime. Then it's additively isomorphic to Z / |a|^2 Z, and the existence of a 6th root of unity in the ring Z / |a|^2 Z gives a way to homomorphically identify it with E / a E (where the sixth roots of unity are the obvious ones).
Then apply the Chinese Remainder Theorem (over Z) to combine all of those prime-power rings together.
-- APG.
Sent: Tuesday, May 21, 2019 at 4:23 AM From: "Dan Asimov" <dasimov@earthlink.net> To: math-fun@mailman.xmission.com Subject: [math-fun] Quotient rings of the ring of Eisenstein integers
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: ----- What is the multiplicative structure? -----
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
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participants (3)
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Adam P. Goucher -
Dan Asimov -
Neil Sloane