### 3 as a Ninth Power (mod p).

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We provide a generalization of Scholz’s reciprocity law using the subfields ${K}_{{2}^{t-1}}$ and ${K}_{{2}^{t}}$ of $\mathbb{Q}\left({\zeta}_{p}\right)$, of degrees ${2}^{t-1}$ and ${2}^{t}$ over $\mathbb{Q}$, respectively. The proof requires a particular choice of primitive element for ${K}_{{2}^{t}}$ over ${K}_{{2}^{t-1}}$ and is based upon the splitting of the cyclotomic polynomial ${\Phi}_{p}\left(x\right)$ over the subfields.

Various multiple Dedekind sums were introduced by B.C.Berndt, L.Carlitz, S.Egami, D.Zagier and A.Bayad.In this paper, noticing the Jacobi form in Bayad [4], the cotangent function in Zagier [23], Egami’s result on cotangent functions [14] and their reciprocity laws, we study a special case of the Jacobi forms in Bayad [4] and deduce a generalization of Egami’s result on cotangent functions and a generalization of Zagier’s result. Further, we consider their reciprocity laws.

We examine primitive roots modulo the Fermat number ${F}_{m}={2}^{{2}^{m}}+1$. We show that an odd integer $n\ge 3$ is a Fermat prime if and only if the set of primitive roots modulo $n$ is equal to the set of quadratic non-residues modulo $n$. This result is extended to primitive roots modulo twice a Fermat number.