Linear magma over the extension field F_81 = F_3[α]/⟨α⁴ + α + 2⟩.
Operation: x ◇ y = a·x + b·y in F_81 with (a, b) = (2α², 1 + α²). Since α + β = 2α² + 1 + α² = 1 + 3α² = 1 in F_81 (char 3), this is the "Type 1" translation-invariant fully-idempotent linear 677 magma, equivalent to x ◇ y = x + β·(y - x) with β = 1 + α² a primitive 10th root of unity in F_81* (which has order 80, divisible by 10).
F_81 is the proper degree-4 extension of F_3; do NOT confuse with the ring Z/81Z (which has zero divisors and is NOT a field).
Size 81, fully idempotent, right-cancellative.
[text written by Claude]
dwrensha · 2026-05-16 11:52:41
Linear magma over the ring Z/81Z: x ◇ y = 21x + 16y (mod 81).
NOTE: Z/81Z is the ring of integers mod 81, NOT a field — since 81 = 3^4 is not prime, Z/81Z has zero divisors (e.g. 3·27 = 0 mod 81). The genuine field F_81 = GF(3^4) exists as a separate object but is not used here; this construction uses ring arithmetic mod 81.
[text written by Claude]
dwrensha · 2026-04-29 17:24:11
Linear magma over Z/81Z (the prime field F_81): x ◇ y = 21x + 16y (mod 81).
dwrensha · 2026-04-29 13:45:46
Linear magma over Z/81Z (the prime field F_81): x ◇ y = (21,16).
dwrensha · 2026-05-16 12:02:52
dwrensha · 2026-05-16 11:52:41
dwrensha · 2026-04-29 17:24:11
dwrensha · 2026-04-29 13:45:46
dwrensha · 2026-04-29 13:29:51