Equation 677 Database

Linear magma over the extension field F_9 = F_3[α]/⟨α² + 1⟩. Operation: x ◇ y = a·x + b·y in F_9 with (a, b) = (1, 2 + α). Note 2 + α is a root of Φ_10(x) = x⁴ - x³ + x² - x + 1 (cyclotomic polynomial for primitive 10th roots of unity) in F_9, and the corresponding α_coef = 1 satisfies α_coef = -β³ - β - 1 - this is Pace Nielsen's "Type 2" linear 677 magma family. F_9 is the proper degree-2 extension of F_3; do NOT confuse with the ring Z/9Z = Z/9 (which has zero divisors and is NOT a field). Size 9, not fully idempotent (only 1 element is idempotent), right-cancellative. [text written by Claude]