Field

A field is an algebraic structure with operations resembling those of $ℚ$ . A field $(𝐾, +, \cdot)$ consists of an abelian group $(𝐾, +)$ with identity $0$ called addition, and an additional abelian group $(𝐾 ∖ {0}, \cdot)$ called multiplication, such that multiplication is distributive over addition #m/def/ring

𝑎 \cdot (𝑏 + 𝑐) = (𝑎 \cdot 𝑏) + (𝑎 \cdot 𝑐)

That is, a field is both a commutative ring and a division ring.

Constructing fields

$𝑅 / 𝐼$ for commutative $𝑅$ is a field iff $𝐼$ is maximal

Properties

$𝑅$ is a field iff it has no nonzero proper ideals
A field contains $ℤ 𝑝$ or $ℚ$

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