[[Field theory MOC]]
# Field

A **field** is an algebraic structure with operations resembling those of [[Rational numbers]].
A field $(K, +, \cdot)$ consists of an [[abelian group]] $(K,+)$ with identity $0$ called **addition**,
and an additional abelian group $(K \setminus \{ 0 \}, \cdot)$ called **multiplication**,
such that multiplication is distributive over addition #m/def/ring
$$
\begin{align*}
a \cdot(b+c) = (a \cdot b) + (a \cdot c)
\end{align*}
$$
That is, a field is both a [[commutative ring]] and a [[division ring]].


## Constructing fields

- [[Condition for a quotient commutative ring to be a field]]

## Properties

- [[Condition for a quotient commutative ring to be a field#^C1]]
- [[A field contains modular arithmetic or the rationals]]

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#state/tidy  | #lang/en | #SemBr