## If a is an integer greater than 1, then aᵈ-1 divides aⁿ-1 if and only if d divides n

Fix an integer . Prove that for all , divides if and only if divides . Conclude that if and only if .

If , then by this previous exercise, divides in , and so divides in .

Conversely, suppose divides . Using the Division Algorithm, say . Now . If , then we have , so that and . If , then , and again we have , so that as desired.

Recall that is precisely the roots (in some splitting field) of . Now if and only if divides by the above argument, if and only if divides by a previous exercise, if and only if divides , if and only if .

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