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*

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