The Supplementary Laws of the Quadratic Reciprocity Law and the Generalized Reciprocity Law

Oswald Baumgart

Chapter Chapter 7 in The Quadratic Reciprocity Law, 2015, pp 71-76 from Springer

Abstract: Abstract In our investigations we have made the assumption that we have already proved the supplementary laws supplementary law of quadratic reciprocity, which can be expressed by the formulas $$\displaystyle{(I)\ \ \Big(\frac{-1} {p} \Big) = (-1)^{\frac{p-1} {2} }\quad \text{and}\quad (\mathit{II})\ \ \Big(\frac{2} {p}\Big) = (-1)^{\frac{p^{2}-1} {8} }.}$$ In this section we will verify formulas (I) and (II) using the methods that we have already used in the chapters above for deriving the relation $$\displaystyle{\Big(\frac{p} {q}\Big)\Big(\frac{q} {p}\Big) = (-1)^{\frac{p-1} {2} \cdot \frac{q-1} {2} }.}$$ First we remark that formula (I) is an immediate consequence of Fermat’s Theorem.

Keywords: Quadratic Residue; Minimal Integer; Quadratic Character; Coprime Integer; Generalize Lemma (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-16283-6_7

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DOI: 10.1007/978-3-319-16283-6_7

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