I am stuck on problem 54 of chapter 13 which states: Let F be afinite field with

Question

I am stuck on problem 54 of chapter 13 which states: Let F be afinite field with n elements. Prove that x^(n-1)=1 for all nonzerox in F.

I know a finite integral domain is a field. An integral domain is acommutative ring with nonzero elements and the cancellationproperty holds. I know I should be focusing on n-1 part of thequestion, which is the number of elements minus 1. A fieldhas unity and every nonzero element is a unit. That means 1 is inthe field b/c it is the multiplicative inverse and each elementthat is not zero has an inverse. I know the characteristic thefield is the order of 1 and that is must be prime. I know a greatdeal about fields but don't understand how to connect the ideas toprove that x^(n-1)=1. I have also tried numerous examples and dobelieve that it is true. Can you please help me (even just a hintwould be greatly appreciated) Thanks!!!

Explanation / Answer

Prove that x^(n-1)=1 for all nonzero x in F. this will be true forall n=1 because all positive numbers to the zero power = 1 and00 is undefined

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