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

Definition §

A field is a set which includes any two unique binary operations known as addition and multiplication. Note that these don’t have to be our usual notion of addition and multiplication, but any binary operators which satisfy the field properties can be used!

General Definition §

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

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

It must satisfy certain properties:

Subfield §

Let $\mathbb{F}$ be a field.

Then, $\mathbb{G}$ is a subfield of $\mathbb{F}$ if:

• The set used to construct $\mathbb{G}$ is a subset of the set used to construct $\mathbb{F}$ i.e. $G\subseteq F$
• $\mathbb{G}$ has the same defined addition and multiplication operations as $\mathbb{F}$ i.e. it borrows the addition and multiplication from $\mathbb{F}$

Resources §

• Abstract Algebra: The definition of a Field- Socratica
• Field Definition (expanded): Abstract Algebram - Socratica
• https://dept.math.lsa.umich.edu/~jchw/2015Math110Material/FieldAxioms-Math110-W2015.pdf