---
tag: AHH97EJ6VH
kind: concept
name: Semifield
msc: [12K10, 17A35]
sources:
  - url: https://en.wikipedia.org/wiki/Semifield
---

## Summary

A semifield is an algebraic structure that satisfies all the axioms of a [[division ring|0EA5BTA0CX]] except possibly the associativity of multiplication.
Semifields arise in finite geometry, where each semifield coordinatizes a projective plane, called a semifield plane.
Finite semifields that are not fields give non-Desarguesian planes.
In other areas, a semifield is a semiring whose nonzero elements form a group under multiplication.
This node is about the first meaning.

## Explanations

### L1 human

In a field one can add, subtract, multiply and divide by any nonzero element, and multiplication is associative and commutative.
A semifield keeps addition, subtraction and division, and the two distributive laws, but drops the rule $(ab)c = a(bc)$.
Every field and every division ring is a semifield.
The octonions, an eight-dimensional algebra over the real numbers, are a semifield whose multiplication is not associative.