Fields

What is it?

A field is a concept used in Mathematics to group numbers with similar characteristics. In theory, it uses a concept called commutative ring, but it goes beyond the scope of this note.

A field is a set of numbers $\mathbb{F}$, with the property that if $a,b\in \mathbb{F}$, then the results of $a+b,a-b,ab$ and $\frac{a}{b}/b\ne 0$, are also inside $\mathbb{F}$.

Fields. Not sets.

While it may seem similar, a field is not the same as a set of numbers. Sets are the popular collection of numbers like $\mathbb{Z},\mathbb{N}$ and $\mathbb{R}$.

However while $3$ and $5$ are natural numbers the result of $3-5$ does not belong to $\mathbb{N}$. This disqualifies the field concept.

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Conditions of a field

A collection of numbers is only a field if, for any possible element, it follows all of the rules:

• Commutativity of addition

  $x+y=y+x$

• Associativity of addition

  $(x+y)+z=(z+y)+x$

• Existence of an additive identity

  There is an element $0\in \mathbb{F}$, called zero, such that $x+0=x$.

• Existence of additive inverses

  For each $x$, there is an element $-x\in \mathbb{F}$, such that $x+(-x)=0$.

• Commutativity of multiplication

  $xy=yx$

• Associativity of multiplication

  $(x\ast y)\ast z=x\ast (y\ast z)$

• Distributivity

  $(x+y)\ast z=x\ast z+y$ and $x\ast (y+z)=x\ast y+z$

• Existence of a multiplicative identity

  There is an element $1\in \mathbb{F}$, such that $1\ne 0$ and $x\ast 1=x$.

• Existence of a multiplicative inverse

  If $x\ne 0$, then there is an element $x^{-1}\in \mathbb{F}$, such that $x\ast x^{-1}=1$.