---
tag: DSE72WQ5A2
kind: concept
name: Near-field
msc: [12K05, 16Y30]
sources:
  - url: https://en.wikipedia.org/wiki/Near-field_(mathematics)
same_as:
  wikidata: Q571787
  wikipedia: Near-field (mathematics)
---

## Summary

A near-field is an algebraic structure like a [[division ring|0EA5BTA0CX]], except that only one of the two distributive laws is required.
It is a near-ring with identity in which every nonzero element has a multiplicative inverse.
Near-fields were first constructed by Dickson, and they coordinatize certain projective planes that are not Desarguesian.

## Explanations

### L1 human

In a field one can add, subtract, multiply and divide, and multiplication distributes over addition from both sides.
A near-field keeps everything except one side of distributivity: $(a + b)c = ac + bc$ always holds, but $a(b + c) = ab + ac$ may fail.
Every field and every division ring is a near-field.
The smallest near-field that is not a division ring has $9$ elements.
It is built from the [[finite field|0ZXZ82FTX2]] with $9$ elements by twisting its multiplication.