Fields and Ordered Fields

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 5 June 2012

Fields and Ordered Fields

A field is a set $𝔽$ with operations $$\begin{array}{rrcl}+:& 𝔽\times 𝔽& ⟶& 𝔽\\ & (a,b)& ⟼& a+b\end{array}\text{and}\begin{array}{rcll}·:& 𝔽\times 𝔽& ⟶& 𝔽\\ & (a,b)& ⟼& a·b=ab\end{array}$$ such that

1. If $a,b,c\in 𝔽$ then $(a+b)+c=a+(b+c)$.
2. If $a,b\in 𝔽$ then $a+b=b+a$.
3. There exists $0\in 𝔽$ such that if $a\in 𝔽$ then $0+a=a+0=a$.
4. If $a\in 𝔽$ then there exists $-a\in 𝔽$ such that $a+\left(-a\right)=\left(-a\right)+a=0$.
5. If $a,b,c\in 𝔽$ then $\left(ab\right)c=a\left(bc\right)$.
6. If $a,b,c\in 𝔽$ then $(a+b)c=ac+bc$ and $c(a+b)=ca+cb$.
7. There exists $1\in 𝔽$ such that if $a\in 𝔽$ then $1·a=a·1=a$.
8. If $a\in 𝔽$ and $a\ne 0$ then there exists $a^{-1}\in 𝔽$ such that $a·a^{-1}=a^{-1}·a=1$.
9. If $a,b\in 𝔽$ then $ab=ba$.

An ordered field is a field $𝔽$ with a total order $\le$ such that

1. If $a,b,c\in 𝔽$ and $a\le b$ then $a+c\le b+c$, and
2. If $a,b\in 𝔽$ and $a\ge 0$ and $b\ge 0$ then $ab\ge 0$,

where $$a<b\text{if}a\le b\text{and}a\ne b,a\ge b\text{if}a\nless b,\text{and}a>b\text{if}a\nleqq b.$$

The absolute value on $𝔽$ is the function $\left|\phantom{n}\right|:𝔽\to {𝔽}_{\ge 0}$ given by $$\left|x\right|=\sup \{x,-x\}.$$

Let $𝔽$ be an ordered field with order $\le$. Then

1. If $a\in 𝔽$ and $a>0$ then $-a<0$.
2. If $a\in 𝔽$ and $a>0$ then $a^{-1}>0$.
3. If $a,b\in 𝔽$ and $a>0$ and $b>0$ then $ab>0$.
4. If $a\in 𝔽$ then $a^2\ge 0$.
5. If $a,b\in 𝔽$ and $a\ge 0$ and $b\ge 0$ then $a\le b$ if and only if $a^2\le b^2$.
6. $1\ge 0$.
7. If $x\ge 0$ and $y\ge 0$ then $x+y\ge 0$.

Notes and References

These fundamental definitions and properties of ordered ordered fields are too often assumed to be true by osmosis. The basic properties of ordered fields appearing in Proposition 1.1??? are used incessantly in working with real numbers.

The definition of ordered fields is given in [Bou, Ch. VI § 2 no. 3 Definition 3]. The definition of absolute value is given in [Bou, Alg. Ch. VI § 1 no. 11 Definition 4]. These definitions also appear in the intorduction to the real numbers given in [Bou, Top. Ch. 4 § 1].

References

[Bou] N. Bourbaki, Algebra, Chapter 6, Masson????? MR?????.

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