F_29² Type-2 linear magma at size 841 (non-fully-idempotent).
Size 841 = 29², NOT fully idempotent (only 1 idempotent: the F_29² zero element), right-cancellative. Linear over the extension field F_29² = F_29[α]/⟨α² − 2⟩.
Operation: x ◇ y = α_coef·x + β·y in F_29², with β a root of Phi_2_5(x) = x⁴ + x³ + 2x² + 2x + 1 and α_coef = -β³ − β − 1. This is Pace Nielsen's "Type 2" construction. Phi_2_5 has 4 roots in F_29², forming 2 Galois-conjugate orbits → 2 distinct iso classes at size 841 (this magma is one Galois orbit; the other is magma#8469938fe79d).
Sub-magma structure: NO size-29 sub-magmas. Every pair of distinct elements generates the full 841-element magma.
Companion to the F_29² Type-1 magmas at size 841: magma#cc0cd58750be and magma#ee91772f0657 (both fully idempotent). The 4 F_29² linear magmas at size 841 (2 Type-1 + 2 Type-2) cover the full Pace Nielsen classification at this size.
The size-841 family also includes magma#ad16fb07 (an exceptional non-standard construction) and other entries — see size-841 page commentary.
[text written by Claude]
dwrensha · 2026-05-20 19:28:51