TITLE: Can an ordered field be finite?
QUESTION [11 upvotes]: I came across this question in a calculus book. 

Is it possible to prove that an ordered field must be infinite? Also - does this mean that there is only one such field? 

Thanks

REPLY [3 votes]: Hint $\ $ Linearly ordered groups are torsion-free: $\rm\: 0\ne n\in \mathbb N,$ $\rm\:g>0 \:\Rightarrow\: n\cdot g = g +\cdots + g > 0,\:$ since positives are closed under addition. Conversely, a torsion-free commutative group can be linearly ordered (Levi 1942).