Fields

Definition

A field is a set $F$ together with two operations, $(+)$ and $(\cdot )$. That is,

$$\begin{array}{rlrl}+& :F\times F\to F& & (a,b)\mapsto a+b\\ \cdot & :F\times F\to F& & (a,b)\mapsto a\cdot b\end{array}$$

These operations satisfy

• Addition axioms
• Multiplication axioms
• Distribution axioms

Ordered Field

An ordered field is a a pair $(F,<)$ where $F$ is a field and $<$ is an ordering such that $F$ becomes an Ordered Set and the following hold:

1. $(\forall a,b,c\in F)(a<b⟹a+c<b+c)$
2. $(\forall a,b\in F)(a,b>0⟹ab>0)$

Example:

• $(\mathbb{Q},+,\cdot ,<)$ is an ordered field.
• $(\mathbb{Z},+,\cdot ,<)$ is not an ordered field. While there is an ordering, the issue is that $\mathbb{Z}$ is not a field because there are no multiplicative inverses.
• Let $p$ be a prime. Then ${\mathbb{F}}_p:=\{0,1,2,\cdots ,p-1\}$ with “clock” addition and multiplication. It is a field, but cannot be ordered. Indeed, since $1\ne 0$ then $1<0$ or $1>0$. Then, $2<1$. Inductively, by adding it $p$ times, we get $0<1$, which is a contradiction on the first case. Case 2 is similar.

Theorem (Completeness of $\mathbb{R}$)

There exists a ordered field which has the least upper bound property. We denote this by $\mathbb{R}$. Moreover, $\mathbb{Q}<\mathbb{R}$. In some places, $\mathbb{R}$ having the least upper bound property is called the Completeness of $\mathbb{R}$.