Field

A field is a number system, or more specifically, a set, which has addition, subtraction, multiplication and division defined in a certain manner. The most common types of fields are the field of real numbers, (and their subfields, like rational and irrational numbers), and the field of complex numbers.

Fields are commonly denoted with $\mathbb{F}$ and the elements belonging to the field are sometimes called scalars

Disclaimer of ignorance

As I’m writing this, I’m an undergrad just beginning my linear algebra course. This stuff might be wrong. A lot of this abstract stuff is weird, and my notation might be very different from what you see on the internet. So this is the disclaimer.

Definition §

A field is a set which includes any two unique Binary Operations. Most commonly, addition and multiplication are used, but any binary operators which satisfy the field properties can be used!

My Definition §

$$\mathbb{F}=(F,\ast ,\star )$$

• $F$ = Set which the field includes
• $\ast ,\star$ = Binary operators. The symbols here are completely arbitrary, but the two operations must be different. I specifically choose the $\ast$ and $\star$ symbol because you’ll never see them anywhere else, so they can be considered the ‘general’ operator symbol

General Definition §

$$\mathbb{F}=(F,+,\times )$$

• $F$ = Set which the field includes
• $+$ = Addition binary operator
• $\times$ = Multiplication binary operator

It must satisfy certain properties:

My Simplified Axioms §

These axioms are more generalised and also simplified. A field can only be defined if it follows these properties. Assume we have elements in the field, $a,b,c\in F$. Also assume that our binary operations are symbolised with $\ast$ and $\star$

1. Closure under operations §

Both the binary operators defined in the field must output a value in the field. This comes from the definition of a binary operator:$$a * b \in F \ \ \ \ \ \ \ \ \ \ \ a\star b \in F

##### 2. Associativity Parenthesis can be rearranged when either of the binary operations are done to elements in a field: $$a* (b*c) = (a*b) * c \ \ \ \ \ \ \ \ a \star(b \star c) = (a\star b) \star c

3. Commutativity §

The two operands for a binary operator can be swapped, and the answer is unaffected:$$ab = ba\ \ \ \ \ \ \ \ \ \ a \star b = b\star a

##### 4. Existence of Identity There must exist an element in $F$ which, if used as input in a binary operation with another element in $F$, does not affect that element. This is the **identity**. $$\text{i} * a = a * \text{i} = a \ \ \ \ \ \ \ \ \ \ {\scr i} \star a = a\star {\scr i} = a

5. Uniqueness of Identity §

The identity for the first binary operator must be different from that of the second binary operator:$$\text{i} \neq {\scr i}

##### 6. Existence of Inverse There must exist an element in $F$ which returns the **identity** if used with another element for the binary operation. This is the **inverse**:$$a * a^{-1} = \text{i} \ \ \ \ \ \ \ \ \ \ a \star a^{-1} = {\scr i}

I’ve chosen to denote the inverse as the superscript (${}^{-1}$) to makes things simpler, but this does not mean the inverse for $\ast$ is also the inverse for $\star$ : $$a * a^{-1} = \text{i} {\centernot \implies} a \star a^{-1} = {\scr i}

### Resources * [Abstract Algebra: The definition of a Field- Socratica](https://www.youtube.com/watch?v=bjp4nF8TW7s) * [Field Definition (expanded): Abstract Algebram - Socratica](https://www.youtube.com/watch?v=KCSZ4QhOw0I&t=405s) * https://dept.math.lsa.umich.edu/~jchw/2015Math110Material/FieldAxioms-Math110-W2015.pdf