Quadratic Reciprocity

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desiresjab
Posts: 4
Joined: Wed Feb 10, 2016 5:16 am UTC

Quadratic Reciprocity

Postby desiresjab » Fri Feb 12, 2016 9:52 am UTC

No one ever called it simple. When something eludes Euler for forty years you can make a wild guess it is pretty deep.

I can state the law, I can calculate the Legendre symbol for any two primes, I can follow Eiesenstein's lattice point proof, I have a handle on Euler's criterion and Gauss's lemma, et al.

Yet that is not enough. I insist on seeing plumb through the law until I can hold it in one thought and do so by peering only at numbers. I do not want any help from groups, subgroups, cosets or symmetries. Maybe I expect to find something that hides behind a tree by shining a flashlight into the forest without going in.

If I were an expert in group theory the answer might pop right out at me in spite of myself--but I am not an expert there. I believe it must be possible to see what I am looking for by the naked light of numbers without resort to higher abstractions.

I look for the actual mechanism, not a proof or a restatement of the law or facts surrounding it. Since it is a law of numbers, it must be possible to see its mechanism through numbers alone.

In particular, I only need to see the mechanism that makes primes of the type 4n+3 behave toward each other the way they do when squared under the other's modulus. If I see this clearly, the rest of the theory for 4n+1 primes and mixed couples will shake right out. In the end, there has to be a mechanical reason which prevents two 4n+3 primes from having the same character.

If you could explain it in the terms I demand, there is no guarantee I would understand you. What I guarantee is the same mighty effort I have put in so far as an amateur in the field.

Cauchy
Posts: 600
Joined: Wed Mar 28, 2007 1:43 pm UTC

Re: Quadratic Reciprocity

Postby Cauchy » Mon Feb 15, 2016 12:28 am UTC

desiresjab wrote:Since it is a law of numbers, it must be possible to see its mechanism through numbers alone.


This is not true for the more advanced parts of number theory. The majority (the vast majority, probably) of semi-recent number theory breakthroughs have involved analyzing the structure of the integers through lenses other than "numbers alone". If you want an example, look no further than Fermat's Last Theorem, a hypothesis which stood unbested for centuries until someone looked at it under the right set of lenses, none of which were remotely close to "numbers alone".

I'm no expert on quadratic reciprocity, so it's possible that someone here will give you the insight you need, but you should ready yourself for the possibility that, like for many true statements about the integers, understanding this particular fact at a deep level will require looking at the integers from a different perspective than you're used to.
(∫|p|2)(∫|q|2) ≥ (∫|pq|)2
Thanks, skeptical scientist, for knowing symbols and giving them to me.

desiresjab
Posts: 4
Joined: Wed Feb 10, 2016 5:16 am UTC

Re: Quadratic Reciprocity

Postby desiresjab » Tue Feb 16, 2016 12:37 am UTC

Thanks for the response--a very worthy response.

I am preparing myself for just the contingency you suggest and for the same suspicions. I thought that by focusing only on the mechanism for 4n+3 primes I might be able to see far enough into the gears, but I am now having serious doubts about that.

Being stubborn has increased my knowledge in and around the subject, so I have no regrets. I learned things I might never have if I had been impatient, and became more familiarized with the unglamorous side of number theory near the beginning of elementary texts on the subject with principles that continually arise later on.

The right abstract algebra text and tutorials seem to be the next step, but I am not absolutely sure. My suspicion is that the behavior of any pair of 4n+3 primes in quadratic reciprocity is due to some kind of asymmetry between sub rings.

I like the numbers themselves, so I would prefer an abstract algebraic or group theoretic approach to number theory as my launching pad, rather than a dry text that never mentions QR and occasionally has numeric examples.

desiresjab
Posts: 4
Joined: Wed Feb 10, 2016 5:16 am UTC

Re: Quadratic Reciprocity

Postby desiresjab » Wed Feb 17, 2016 7:58 am UTC

Well, if any one cares, it turned out, the mechanism behind quadratic reciprocity was visible from ground level. I finally saw it. It is embarrassing, because I have been staring at it all along. Math does not come natural to me. It takes a long time for all the pieces I am staring at to fall into place so I can gain ituitive understanding of the mechanical process inside the machine.

QR is perhaps one of the last laws visible from a ground level view of numbers in modern number theory. To move higher comfortably will require an intimate knowledge of abstract algebra. I am lucky enough to already know a bit about groups, rings and fields et al. But being able to read their notations and being able to work in them are two different propositions.

Anyway, when I saw the reason for the behavior of 4n+3 primes under QR, as I had suspected, the reason for the behavior of 4n+1 primes and mixed couples was immediately illuminated, as well.


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