Hyperfield of a field quotient

Core idea

Let $K$ be a field and $G\leq K^\times$ a multiplicative subgroup. The orbit set

K/G=\{0\}\sqcup K^\times/G

is a quotient hyperfield with

[a][b]=[ab],\qquad [a]\boxplus[b]=\{[ag+bh]:g,h\in G\}.

It is the hyperfield specialization of the quotient-hyperring construction.

Why this is a hyperfield

The formula is independent of the representatives because changing $a$ or $b$ only changes the allowed factors from $G$ . Every nonzero orbit has inverse $[a^{-1}]$ , and distributivity follows from distributivity in $K$ . The entire set of possible orbits is retained; no representative of a hyper-sum is chosen.

The orbit map

\pi:K\longrightarrow K/G

is a weak hyperfield homomorphism and is generally not strong. A fixed sum $a+b$ has one orbit, while independently rescaling the two summands can produce several orbits in $[a]\boxplus[b]$ .

Standard examples

$G=\{1\}$ recovers the original field with singleton-valued addition.
If $|K|\geq3$ , then $K/K^\times$ is the Krasner hyperfield. For $K=\mathbb F_2$ , the same orbit construction recovers the ordinary field $\mathbb F_2$ , so the size hypothesis is real.
$\mathbb R/\mathbb R_{>0}$ is the sign hyperfield.
$\mathbb C/\mathbb R_{>0}$ is the phase hyperfield.

Not every hyperfield is a field quotient

The quotient construction supplies many central examples, but it is not a classification of hyperfields. Massouros constructed hyperfields not isomorphic to $K/G$ for any field $K$ and multiplicative subgroup $G$ . Consequently “hyperfield” and “quotient hyperfield” are not synonyms.

References

Alain Connes and Caterina Consani, “The hyperring of adèle classes,” Journal of Number Theory 131 (2011), 159–194. arXiv:1001.4260. Relevant: quotient hyperrings and hyperfields.
Matthew Baker and Nathan Bowler, “Matroids over partial hyperstructures,” Advances in Mathematics 343 (2019), 821–863. arXiv:1709.09707. Relevant: Remark 2.7 and the standard quotient examples.
Christos G. Massouros, “Methods of constructing hyperfields,” International Journal of Mathematics and Mathematical Sciences 8 (1985), 725–728. EuDML record and text. Relevant: non-quotient hyperfields.