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 §

Field

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!

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

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

It must satisfy certain properties, known as the field axioms:

Field Axioms

Warning $\mathbb{F}$ is not actual notation

I just tend to use it instead of writing $(F,+,\times )$ all the time. The $\mathbb{F}$ generally refers to ${\mathbb{F}}_2$

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 $\mathbb{F}$

Field Examples §

• $\mathbb{Q}$ is a field
• $\mathbb{Z}$ is not a field, because it does not satisfy the axiom M5. Multiplicative Inverses, since the multiplicative inverse of an integer, for example, 2, is not an integer.
• $\mathbb{R}$ is a field
• We can have finite fields ${\mathbb{F}}_2$#todo

The finite Field ${\mathbb{F}}_2$ can be defined with $F=0,1$, with $0_F=0$ and $1_F=1$. #TODO Show how addition and multiplication satisfies the axioms

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