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Prove that there is no field of order 6.

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Prove that there is no field of order 6.
(You can not refer to the fact that all finite field are of prime-power order)

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Solution Summary

It is proven that there is no field of order 6.

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Proof:
Let F be an fields with order |F|=6. Then we consider its multiplicative group F*=F-{0}.
We know that |F*|=6-1=5 is a prime number and thus F* must be a cyclic group.
We can suppose ...

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