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Published in:
2015 | OriginalPaper | Chapter
11. Eisenstein’s Proofs Using Complex Analysis
Author : Oswald Baumgart
Published in: The Quadratic Reciprocity Law
Publisher: Springer International Publishing
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Abstract
If r represents a half-system modulo q, then so does rp. Setting \(\mathit{rp} \equiv \varepsilon r^{{\prime}}\bmod q\), where \(\varepsilon = \pm 1\), and where r
′
belongs to the same half-system as r, then for an arbitrary integer ω we have This implies where P denotes any simply periodic function with period ω.
$$\displaystyle{\frac{\mathit{pr}\omega } {q} \equiv \frac{\varepsilon r^{{\prime}}\omega } {q} \bmod \omega.}$$
$$\displaystyle{P\Big(\frac{\mathit{pr}\omega } {q} \Big) = P\Big(\frac{\varepsilon r^{{\prime}}\omega } {q} \Big),}$$