Fields

Axioms

A field is a set $\mathbb{F}$ with two operations, called addition and multiplication, which satisfy the following field axioms:

• Axioms for addition
  • Closure: $x\in \mathbb{F}$ and $y\in \mathbb{F}$ $⟹$ $x+y\in \mathbb{F}$.
  • Associativity: $(x+y)+z=x+(y+z)$ for all $x,y,z\in \mathbb{F}$.
  • Identity: $\mathbb{F}$ contains an element $0$ such that $0+x=x$ for all $x\in \mathbb{F}$.
  • Inverse: To every $x\in \mathbb{F}$ corresponds an element $-x\in \mathbb{F}$ such that $x+(-x)=0$.
  • Commutativity: $x+y=y+x$ for all $x,y\in \mathbb{F}$.
• Axioms for multiplication
  • Closure: If $x\in \mathbb{F}$ and $y\in F$, then their product $xy\in \mathbb{F}$.
  • Associativity: $(xy)z=x(yz)$ for all $x,y,z\in \mathbb{F}$.
  • Identity: $\mathbb{F}$ contains an element $1\ne 0$ such that $1x=x$ for every $x\in \mathbb{F}$.
  • Inverse: If $x\in \mathbb{F}$ and $x\ne 0$ then there exists an element $\frac{1}{x}\in \mathbb{F}$ such that $x(\frac{1}{x})=1$.
  • Commutativity: $xy=yx$ for all $x,y\in \mathbb{F}$.
• Distributive Law
  • $x(y+z)=xy+xz$ holds for all $x,y,z\in \mathbb{F}$.

Statements implied by the axioms

• For addition
  • if $x+y=x+z$ then $y=z$
  • if $x+y=x$ then $y=0$.
  • If $x+y=0$ then $y=-x$.
  • $-(-x)=x$.
• For multiplication
  • If $x\ne 0$ and $xy=xz$ then $y=z$.
  • If $x\ne 0$ and $xy=x$ then $y=1$.
  • If $x\ne 0$ and $xy=1$ then $y=\frac{1}{x}$.
  • If $x\ne 0$ then $\frac{1}{\frac{1}{x}}=x$.
• And the rest
  • $0x=0$.
  • If $x\ne 0$ and $y\ne 0$ then $xy\ne 0$.
  • $(-x)y=-(xy)=x(-y)$.
  • $(-x)(-y)=xy$.

Examples

• The set $\{0,1,2,3,4\}$ is a field when addition and multiplication are defined modulo $5$. Pay special attention to how multiplicative inverses would work here. $4^{-1}=4,3^{-1}=2,2^{-1}=3,1^{-1}=1$.