91Ó°ÊÓ

If \(p=2^{n}+1\) is a Fermat prime, show that 3 is a primitive root modulo \(p\).

Show that the sum of all the primitive roots modulo \(p\) is congruent to \(\mu(p-1)\) modulo \(p\).

Let \(A\) be a finite abelian group and \(a, b \in A\) elements of order \(m\) and \(n\), respectively. If \((m, n)=1\). prove that \(a b\) has order \(m n\).

Use the fact that 2 is a primitive root modulo 29 to find the seven solutions to \(x^{7} \equiv 1(29)\)

Let \(p\) be a prime and \(d\) a divisor of \(p-1\). Show that the \(d\) th powers form a subgroup of \(U(Z / p \mathbb{Z})\) of order \((p-1) / d\). Calculate this subgroup for \(p=11, d=5 ; p=17\), \(d=4 ; p=19, d=6\)