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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 ℚ ( ζ p ) , of degrees 2 t - 1 and 2 t over 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 Φ p ( x ) over the subfields.

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