Linear magma over the extension field F_49 = F_7[α]/⟨α² - 3⟩.
Operation: x ◇ y = a·x + b·y in F_49 with (a, b) = (2, 1 + α). Here β = 1 + α is a root of Φ_2_5(x) = x⁴ + x³ + 2x² + 2x + 1 in F_49, and α_coef = -β³ - β - 1 = 2 (in F_49). This is Pace Nielsen's Type-2 non-fully-idempotent linear 677 magma family.
F_49 is the proper degree-2 extension of F_7; do NOT confuse with the ring Z/49Z (which has zero divisors and is NOT a field). Note that the size-49 family in the DB includes two distinct constructions: this one (over F_49) and others (magma#bd9c4bf0, magma#a9267aad) over the ring Z/49Z.
Size 49, not fully idempotent, right-cancellative.
[text written by Claude]
dwrensha · 2026-05-16 11:52:40
Linear magma over the ring Z/49Z: x ◇ y = 2x + 14y (mod 49).
NOTE: Z/49Z is the ring of integers mod 49, NOT a field — since 49 = 7^2 is not prime, Z/49Z has zero divisors (e.g. 7·7 = 0 mod 49). The genuine field F_49 = GF(7^2) exists as a separate object but is not used here; this construction uses ring arithmetic mod 49.
[text written by Claude]
dwrensha · 2026-04-29 17:24:10
Linear magma over Z/49Z (the prime field F_49): x ◇ y = 2x + 14y (mod 49).
dwrensha · 2026-04-29 13:45:46
Linear magma over Z/49Z (the prime field F_49): x ◇ y = (2,14).
dwrensha · 2026-05-16 12:02:52
dwrensha · 2026-05-16 11:52:40
dwrensha · 2026-04-29 17:24:10
dwrensha · 2026-04-29 13:45:46
dwrensha · 2026-04-29 13:29:51