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 $(F,\oplus,\odot)$ | $\oplus$ ✔, $\odot$ ✔ |
$\oplus$ ✔, $\odot$ ✔ |
$\oplus$ ✔, $\odot$ ✔ |
$\oplus$ ✔, $\odot$ ✔ |
$\oplus$ ✔, $\odot$ ✔ |
$\oplus$ ✔, $\odot$ ✔ |
✔ |
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| created: 2019-08-09 19:30:16 | modified: 2019-08-09 19:30:16 | by: bookofproofs
