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A generalization of Scholz’s reciprocity law

Mark Budden, Jeremiah Eisenmenger, Jonathan Kish (2007)

Journal de Théorie des Nombres de Bordeaux

We provide a generalization of Scholz’s reciprocity law using the subfields K 2 t - 1 and K 2 t of ( ζ p ) , of degrees 2 t - 1 and 2 t over , 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 Φ p ( x ) over the subfields.

A generalization of the reciprocity law of multiple Dedekind sums

Masahiro Asano (2007)

Annales de l’institut Fourier

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.

A necessary and sufficient condition for the primality of Fermat numbers

Michal Křížek, Lawrence Somer (2001)

Mathematica Bohemica

We examine primitive roots modulo the Fermat number F m = 2 2 m + 1 . We show that an odd integer n 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.

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