There are exactly 178 Krasner hyperfields of order eight, and which hyperfields of order at most eight are quotients of finite fields
Abstract
A Krasner hyperfield is a field-like structure whose addition is multivalued. Let Hₙ denote the number of isomorphism classes of hyperfields of order n. The values H₂, …, H₆ = 2, 5, 7, 27, 16 are due to Ameri, Eyvazi and Hošková-Mayerová and H₇ = 277 to Massouros and Massouros. We compute the next value: H₈ = 178. The lower bound is a list of 178 pairwise non-isomorphic tables; the upper bound is an exhaustive search over all 128⁴ = 268 435 456 tables admitted by an elementary reduction, which is stated and proved by hand in full below. In the notation of the On-Line Encyclopedia of Integer Sequences this is A343596(8) = 178, the published entry stopping at order 6. We then determine, for every order n with 3 ≤ n ≤ 8, exactly which of the Hₙ isomorphism classes are quotients 𝔽_q/G of a finite field by a subgroup of its multiplicative group. The counts are 4 of 5, 4 of 7, 9 of 27, 7 of 16, 15 of 277 and 9 of 178 at orders 3, 4, 5, 6, 7, 8; the first two reproduce a theorem of Baker and Jin. The order-5 case answers the finite-field half of Baker and Jin's open question (1): the nine quotient classes all have multiplicative group C₄, and none of the eleven classes with multiplicative group C₂ × C₂ is a quotient of a finite field. The infinite-field half of their question is untouched and remains open. The counts 9, 7, 15 at orders 5, 6, 7 agree with counts obtained by Linzi by a different route; what is new for those orders is the identification of which classes they are. The numbered results below are machine-checked in Lean 4. Section [sec:verif] says which of them rest on exhaustive computation, how large each search is, and which steps of the argument are made by hand rather than by machine.
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Stated results
KT1known data2026-09-03
The complete census of Krasner hyperfields of orders 3 to 7 as explicit tables: 5, 7, 27 (= 16 over C₄ plus 11 over C₂ x C₂), 16 and 277 isomorphism classes, each list verified to consist of hyperfields, to be in canonical form under the automorphism group of the multiplicative group, and to be pairwise non-isomorphic
KT2candidate2026-09-03
There are exactly 178 isomorphism classes of Krasner hyperfields of order 8 (equivalently A343596(8) = 178): 178 explicit pairwise non-isomorphic tables, and an exhaustive search over all 128⁴ = 268435456 admissible presentations showing there are no others
KT3known data2026-09-03
The finite-field quotient classes at orders 3 and 4, located inside the census: exactly 4 of the 5 hyperfields of order 3 and exactly 4 of the 7 of order 4 are quotients F_q/Gᵣ, each realised by an explicit finite field whose irreducibility and primitivity certificate is checked in Lean
KT4candidate2026-09-03
Baker-Jin open question (1), finite-field half: of the 27 isomorphism classes of hyperfields of order 5, exactly 9 are quotients of finite fields and 18 are not – the 9 sit among the 16 classes with multiplicative group C₄, and none of the 11 classes with multiplicative group C₂ x C₂ is a quotient, because F_qˣ/G is cyclic
KT5candidate2026-09-03
The same matching at orders 6, 7 and 8: exactly 7 of the 16 hyperfields of order 6, exactly 15 of the 277 of order 7, and exactly 9 of the 178 of order 8 are quotients of finite fields
KT6routine2026-09-03
Negative controls for the census and the quotient matching: a table failing the axioms outright, a commutative reversible table with unique inverses that fails associativity, a 179th order-8 class refuted, a 177-class census shown incomplete, a reducible modulus rejected by the field certificate, and C₂ x C₂ shown not isomorphic to C₄
KT7known data2026-09-03
Independent reproduction of both printed tables of arXiv:2608.03625: Nᵣᵉmp = 6, 17, 42, 102, 278, 492, 762 with 2, 4, 7, 7, 18, 8, 18 exceptional prime powers for r = 2..8, and Qᵣᶠin = 2, 4, 4, 9, 7, 15 for r = 1..6
This ledger entry is reported in prose and is not bound to a Lean theorem.KT8known data2026-09-03
Independent confirmation of the values the source's companion repository added after the arXiv v1: Nᵣᵉmp = 578, 1182, 1410, 1742 with 15, 27, 11, 41 exceptions for r = 9, 10, 11, 12, and Qᵣᶠin = 9, 17, 14, 27, 9, 36 for r = 7..12
This ledger entry is reported in prose and is not bound to a Lean theorem.KT9measurement2026-09-03
Cost and collision measurements: the order-8 census is a 2.7 x 10⁸-candidate search that a proved single reversibility instance cuts 519-fold to 777000 nodes; the order-9 census (multiplicative group C₈) is 1.3 x 10¹1 candidates and is parked
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
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- Machina Mathematica
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- Korea Superintelligence Labs
- Source context
- A hyperfield (Krasner) is (F, ⊞, ·, 0, 1) where (F 0, ·) is an abelian group, (F, ⊞, 0) is a *canonical hypergroup* — a commutative, associative, reversible hyperoperation F × F → P*(F) with 0 as identity and unique additive inverses — multiplication distributes over ⊞, and 0 is absorbing. This is Definition 1 of arXiv:2608.03625, following Krasner (1957, 1983).
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