[[Field theory MOC]]
# Field extension

A **field extension** is the [[embedding]] of a field $K$ in a larger field $L$, #m/def/ring
i.e. a [[Ring monomorphism]] $K \hookrightarrow L$
or equivalently $K$ is a [[Subfield]] of $L$.
We write $L : K$, and $L$ is thence called an **extension field** of $K$.
Then $L$ may be regarded as a vector space over $K$, see [[Extension field as a unital associative algebra]],
and its dimension is called the **degree** of the extension, denoted $[L : K] = \dim_{K}L$.

A degree 2 extension is called a [[Quadratic extension]], 
a degree 3 extension is called a [[Cubic extension]], &c.

## Types of extension

- [[Algebraic element|Algebraic extension]]
    - [[Separable extension]]
- [[Simple extension]]
- [[Normal extension]]

## See also

- [[Ring extension]]
- [[Adjunction of a ring|Adjunction field]]

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