Fields and Ordered Fields

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

and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 8 February 2010

Fields and ordered fields

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

1. If $a,b,c\in 𝔽$ then $\left(a+b\right)+c=a+\left(b+c\right)$.
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 $\left(a+b\right)c=ac+bc$ and $c\left(a+b\right)=ca+cb$.
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 the following conditions are satisfied.

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

Where we write

1. $a<b$ if $a\le b$ and $a\ne b$,
2. $a\ge b$ if $a\nless b$, and
3. $a>b$ if $a\nleqq b$.

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

Let $𝔽$ be an ordered field with order $\le$. Then the following statements hold.

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 $x+y\ge 0$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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