# Field Morphisms

module Algebra.fieldMorphisms where

open import Agda.Primitive using (Level; __; lsuc; lzero)
open import Types.relations
open import Types.equality
open import Types.functions2

open import Algebra.groups
open import Algebra.groups2
open import Algebra.groupMorphisms
open import Algebra.ringMorphisms

open import Algebra.fields
open import Algebra.fields2

We keep following the same pattern as we have been doing for groups and rings. A morphisms between two fields is a structure-preserving map in a way that preserves the field properties.

## Field Homomorphism

A field homomorphism between two fields 𝔸 and 𝔹 is a function

f : 𝔸 → 𝔹

such that ∀ x, y in 𝔸,

1. f is congruent over +, or $f(x + y) = f(x) + f(y)$
2. f is congruent over , or $f( x ⋅ y) = f(x) ⋅ f(y)$
3. Identities are preserved, i.e. $f(1) = 1$

The fields are isomorphic if f is bijective.

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