Mathematical Field

==2024-12-11

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Field is not always in football.

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Theory

A field in mathematics is a Set $\mathbb{F}$ equipped with two binary operations, typically referred to as addition $+$ and multiplication $\ast$, satisfying specific properties.

The formal definition can be outlined as follows: Field

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Informal Defination:

1. A field consists of a set $\mathbb{F}$ containing at least two elements.
2. Binary Operations: The operations + and ⋅ must be defined for all elements in $\mathbb{F}$.
3. Axioms: The following axioms must hold for all elements $a,b,c\in F$:
   1. Closure: For all $a,b\in \mathbb{F}$, both a+b and a.b are in $\mathbb{F}$
   2. Associativity(Both multiplication and addition)
   3. Commutativity(Both multiplication and addition)
   4. Existence of identity.
   5. Existence of inverse.
   6. Distributive Property

Notation

A field $\mathbb{F}$ is simply denoted as so, $\mathbb{F}(F,+,.)$

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Examples

A = {1,0} is a field. $\mathbb{R}$ is a field.

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PTR

1. Field is essentially combination of two Mathematical Groups.
2. Field is the triplet of set, addition operation and multiplication.

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Sources

1. The Field Axioms