Field extension

Definition

Let $K$ and $L$ be fields. The field $L$ is an extension of $K$ if $K$ is a subfield of $L$. One writes $L/K$ for this field extension.

Remarks

Often one also calls $L$ an extension of $K$ when there is an injective ring homomorphism $K \to L$. Then one identifies $K$ with its image. An extension $L/K$ makes $L$ a vector space over $K$. Its dimension is the degree of the extension. Examples: $\mathbb{C}/\mathbb{R}$, $\mathbb{R}/\mathbb{Q}$ and $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$. The notation $L/K$ does not mean a quotient.

Explanations

L1 human

A field extension is a bigger field that contains a given field. For example, the complex numbers extend the real numbers by an element $i$ with $i^2 = -1$. Field theory studies how the bigger field is built from the smaller one.