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Friday, October 21, 2011

Quadratic reciprocity in a finite group.

Law of Quadratic Reciprocity

Let $p$ and $q$ be distinct odd primes. Then $$\displaystyle \left({\frac p q}\right) \left({\frac q p}\right) = \left({-1}\right)^{\frac {\left({p-1}\right) \left({q-1}\right)} 4}$$ where $\displaystyle \left({\frac p q}\right)$ and $\displaystyle \left({\frac q p}\right)$ are defined as the Legendre Symbol $\displaystyle \left({\frac{a}{p}}\right) := a^{\frac{(p-1)}{2}} \pmod p$.

Gauss considered his work on the Quadratic Reciprocity Law among his major achievements. I don't 'get that', not now anyway, that's a call for more study on the topic.

Now and then, when I browse through papers, or otherwise, I find an interesting mathematical paper... ( that I can actually read ). Actually, I was browsing through a book called Reciprocity Laws, from Euler to Eisenstein by Franz Lemmermeyer, it contains more than 100 proofs of the Quadratic Reciprocity Law. I hoped to find a proof I could appreciate by it's beauty. Although most proofs are based on Gauss's Lemma ( as the proof in M381 ) but there are proofs in other realms of mathematics like Group Theory. Group Theory -as we know it today- did not exist in Gauss's time. That's why I am going to spend some time studying the following paper 'Quadratic reciprocity in a finite group.'

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