field extension

Definition

A field extension is a non-zero homomorphism of fields

$$k⟶K.$$

The field $K$ is called an extension of $k$.

Non-zero field homomorphims

We can think of $k\subset K$ since any non-zero ring homomorphism

f:k \to R

is injective since $\ker f \subset k$ is an ideal of $k$, but the only ideals of $k$ is $(0)$ or $k$ because it is a field.

Degree of extension

The degree of an extension $k\subset K$ is

$$[K:k]:={\dim }_{k}K\in {\mathbb{Z}}_{\ge 1}\cup \{\infty \}$$

Examples

• For $\mathbb{Q}\subset \mathbb{R}$, $[\mathbb{R}:\mathbb{Q}]=\infty$ by cardinality of $\mathbb{Q}$ versus $\mathbb{R}$.

• For $\mathbb{R}\subset \mathbb{C}$, $[\mathbb{C},\mathbb{R}]=2$ since a basis for $\mathbb{C}$ over $\mathbb{R}$ is $\{1,i\}$.

Simple extensions

An extension $k\subset K$ is called simple if $K=k(\alpha )$ where $k(\alpha )$ is the smallest subfield of $K$ which contains $k$ and $\alpha$.

Field characteristic and extensions

Let $k\subset K$ be an extension. Then $k$ and $K$ have equal characteristic.

Proof

Since an extension is a homomorphism of fields, then $1\mapsto 1$, so the characteristic will be the same.