Glossary of Field Theory - Homomorphisms

Homomorphisms

Field homomorphism

A field homomorphism between two fields E and F is a function

f : E → F

such that

f(x + y) = f(x) + f(y)

and

f(xy) = f(x) f(y)

for all x, y in E, as well as f(1) = 1. These properties imply that f(0) = 0, f(x-1) = f(x)-1 for x in E with x ≠ 0, and that f is injective. Fields, together with these homomorphisms, form a category. Two fields E and F are called isomorphic if there exists a bijective homomorphism

f : E → F.

The two fields are then identical for all practical purposes; however, not necessarily in a unique way. See, for example, complex conjugation.