---
tag: EZVN87C88J
kind: concept
name: Field homomorphism
msc: [12F05, "12E99"]
specializes: [0VTH8KPCG2]
sources:
  - url: https://en.wikipedia.org/wiki/Field_(mathematics)
---

## Summary

A field homomorphism is a map between two fields that respects addition, multiplication and the unit element.
It is a [[ring homomorphism|0VTH8KPCG2]] whose source and target are fields.
Every field homomorphism is injective, so it identifies the first field with a subfield of the second.
This is why field theory speaks of embeddings and of [[field extensions|0EQWVHF3W5]].

## Explanations

### L1 human

A field homomorphism copies one field into another without breaking any computation.
The inclusion of the rational numbers into the real numbers is one.
Complex conjugation $a + bi \mapsto a - bi$ is a field homomorphism from $\C$ to itself.
Such a map can never send a nonzero number to $0$, so it never merges two elements.
The [[definition|JBZMRZY4YF]] lists the conditions.