field extension

Definition

A field extension is a non-zero homomorphism of fields

The field is called an extension of .

Non-zero field homomorphims

We can think of since any non-zero ring homomorphism

is injective since is an ideal of , but the only ideals of is or because it is a field.

Field characteristic and extensions

Let be an extension. Then and have equal characteristic.

Proof

Since an extension is a homomorphism of fields, then , so the characteristic will be the same.