Proof
(related to Theorem: Second Supplementary Law to the Quadratic Reciprocity Law)
- By hypothesis, $p > 2$ is an odd prime number.
- In the proof of the quadratic reciprocity law, we have derived the congruence.
$$\frac{p^2-1}8(q-1)\equiv \sum_{k=1}^{\frac{p-1}{2}}q_k+m\mod 2.$$
- For $q=2$, we have $\frac{p^2-1}8\equiv m\mod 2.$
- Therefore, according to the Gaussian lemma, the following formula for the Legendre symbol follows:
$$\left(\frac 2p\right)=(-1)^m=(-1)^{\frac{p^2-1}8}.$$
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References
Bibliography
- Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927
Footnotes
