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Let p be a prime and let a be a primitive root modulo p. Show that...

Let p be a prime and let a be a primitive root modulo p. Show that if gcd (k, p-1) = 1, then b≡ak (mod p) is also a primitive root modulo p.

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Answer #1

Since is a primitive root of unity modulo prime , there exists such that , and if . Note that implies . Suppose that . Then there exists integers such that . If then

Thus, is a root of unity modulo . If possible, suppose that there is such that . Then we get

From we get ; therefore,

Since and , this implies . Also, from Fermat's little theorem we get . Therefore,

But this contradicts the assumption of primitivity of since (as we explained at the beginning) primitivity implies if . Therefore, we have proved that is a primitive -th root of unity modulo .

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