Field theory MOC

Field

A field is an algebraic structure with operations resembling those of Rational numbers. A field (𝐾, +, ⋅) consists of an abelian group (𝐾, +) with identity 0 called addition, and an additional abelian group (𝐾 ∖{0}, ⋅) called multiplication, such that multiplication is distributive over addition ring

𝑎⋅(𝑏+𝑐)=(𝑎⋅𝑏)+(𝑎⋅𝑐)

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

• ^C1
• A field contains modular arithmetic or the rationals

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