Let be a field and with Prove that is the solution set of the equation Remark: In general, for any integer and any of order the solution set of the equation is still (David Hilbert)
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Finally I got it! Let be the solution set. Clearly ; we want to show . Let ; suppose . Clearly then ; but we have hence ; i.e. , and clearly is closed under taking inverses so . To show (clearly ), take any such that ; then hence .
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