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Question: Let F be a field, that is, a set with operations $+$ and $\cdot$ which satisfy the axioms of the definition of an "ordered field". Prove that if $F$ is finite (i.e. has only finitely many elements), then there does not exist an ordering $<$ which makes $F$ into an ordered field.

How can I go about proving a finite field has no order?

David
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Brittany
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1 Answers1

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HINT: There is a positive integer $n$ such that $\underbrace{1_F+1_F+\ldots+1_F}_{n\text{ copies}}=-1_F$.

Brian M. Scott
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