Linear magma over the extension field F_16 = F_2[α]/⟨α⁴ + α + 1⟩.
Operation: x ◇ y = a·x + b·y in F_16 with (a, b) = (1 + α + α², 1 + α³). Here β = 1 + α³ satisfies the "Type 2" cyclotomic condition β⁴ + β³ + 1 = 0 in characteristic 2 (= β⁴ + β³ + 2β² + 2β + 1 with 2 = 0), and α_coef = -β³ - β - 1 = 1 + α + α². This is Pace Nielsen's Type-2 non-fully-idempotent linear 677 magma family.
F_16 is the proper degree-4 extension of F_2; do NOT confuse with the ring Z/16Z (which has zero divisors and is NOT a field).
Size 16, not fully idempotent, right-cancellative. Compare with magma#6fa95655, the F_16 Type-1 (fully idempotent) variant.
[text written by Claude]
dwrensha · 2026-05-16 11:52:40
Linear magma over the ring Z/16Z: x ◇ y = 6x + 11y (mod 16).
NOTE: Z/16Z is the ring of integers mod 16, NOT a field — since 16 = 2^4 is not prime, Z/16Z has zero divisors (e.g. 2·8 = 0 mod 16). The genuine field F_16 = GF(2^4) exists as a separate object but is not used here; this construction uses ring arithmetic mod 16.
[text written by Claude]
dwrensha · 2026-04-29 17:23:41
Linear magma over Z/16Z (the prime field F_16): x ◇ y = 6x + 11y (mod 16).
dwrensha · 2026-04-29 13:44:58
Linear magma over Z/16Z (the prime field F_16): x ◇ y = (6,11).
dwrensha · 2026-05-16 12:02:52
dwrensha · 2026-05-16 11:52:40
dwrensha · 2026-04-29 17:23:41
dwrensha · 2026-04-29 13:44:58
dwrensha · 2026-04-29 13:29:22