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Fields (Overview)

In this chapter, we will introduce the field as a very important algebraic structure. Like it was the case in rings, a field has two different binary operations, usually referred to as “addition” and “multiplication”. The following tabular overview indicates the properties of a field in comparison to previous algebraic structures:

Algebra $(X,\ast)$ Closure Associativity Neutral Element Existence of Inverse Cancellation Commutativity Distributivity
Magma (✔) (✔) (✔) (✔) (✔) n/a
Semigroup (✔) (✔) (✔) (✔) n/a
Monoid (✔) (✔) (✔) n/a
Group (✔) n/a
Ring $(R,\oplus,\odot)$ $\oplus$ ✔,
$\odot$ ✔
$\oplus$ ✔,
$\odot$ ✔
$\oplus$ ✔,
$\odot$ (✔)
$\oplus$ ✔,
$\odot$(✔)
$\oplus$ ✔,
$\odot$ (✔)
$\oplus$ ✔,
$\odot$ (✔)
Field $(F,\oplus,\odot)$ $\oplus$ ✔,
$\odot$ ✔
$\oplus$ ✔,
$\odot$ ✔
$\oplus$ ✔,
$\odot$ ✔
$\oplus$ ✔,
$\odot$ ✔
$\oplus$ ✔,
$\odot$ ✔
$\oplus$ ✔,
$\odot$ ✔

| | | | created: 2019-08-09 19:30:16 | modified: 2019-08-09 19:30:16 | by: bookofproofs

1.Definition: Field

2.Definition: Subfield

3.Definition: Field Homomorphism

Edit or AddNotationAxiomatic Method

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Bibliography (further reading)