Field

A Principle Ideal Domain that has a Multiplicative Inverse.

Definition

A field is a set $F$ that has closure under:

• Addition
• Multiplication With axioms of

1. Additive Identity ($\exists 0_{\mathbb{F}}\text{ s.t }\forall a\in \mathbb{F},0_{\mathbb{F}}+a=a$)
2. Additive Inverse ($\forall a\in \mathbb{F},\exists -a\in \mathbb{F}\text{ s.t }(a)+(-a)=0_{\mathbb{F}}$)
3. Associativity of Addition ($\forall a,b,c\in \mathbb{F},(a+b)+c=a+(b+c)$)
4. Commutivity of Addition $\forall a,b\in \mathbb{F},a+b=b+a$
5. Multiplicative Identity ($\exists 1_{\mathbb{F}}\in \mathbb{F}\text{ s.t }\forall a\in \mathbb{F},1_{\mathbb{F}}\ast a=a$)
6. Multiplicative Inverse ($\forall a\in \mathbb{F},\exists \frac{1}{a}$ s.t $a\ast \frac{1}{a}=1$)
7. Associative with Multiplication $\forall a,b,c\in \mathbb{F},a{x}_{\mathbb{F}}(b{x}_{\mathbb{F}}c)=(a{x}_{\mathbb{F}}b)x_{\mathbb{F}}c$
8. Commutivity of Multiplication $\forall a,b\in \mathbb{F},a{x}_{\mathbb{F}}=b{x}_{\mathbb{F}}a$
9. Distributivity over Addition $\forall a,b,c\in \mathbb{F},a{x}_{\mathbb{F}}(b{+}_{\mathbb{R}}c)+(a{x}_{\mathbb{F}}b){+}_{\mathbb{F}}(a{x}_{\mathbb{F}}c)$

Uniqueness Field Properties Definition

Examples

• Real Number
• Rational Number
• Complex Number (As its a superset of Real Number)
• Integer Modulo n (if $n$ is Prime Number)

Non-Examples

• Natural Number (No additive inverses)
• Integers (No multiplicative inverses)

Concepts

• Uniqueness Field Properties
• Subfield