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 2019-05-01, 10:12 #1 bhelmes     Mar 2016 5·59 Posts quadratic reciprocity law A peaceful day for all members, Does there exist only one quadratic reciprocity law, or do there exist several quadrac reciprocity laws depending on the quadratic polyonoms, resp. is the quadratic reciprocity law the same for gaussian and eisenstein primes. https://en.wikipedia.org/wiki/Reciprocity_law https://en.wikipedia.org/wiki/Gaussi...aussian_primes https://en.wikipedia.org/wiki/Eisenstein_prime Thanks in advance for a clear answer, Bernhard
 2019-05-01, 11:06 #2 GP2     Sep 2003 32×7×41 Posts Why not link as well: https://en.wikipedia.org/wiki/Quadratic_reciprocity
2019-05-01, 13:59   #3
Dr Sardonicus

Feb 2017
Nowhere

101028 Posts

Quote:
 Originally Posted by bhelmes A peaceful day for all members, Does there exist only one quadratic reciprocity law, or do there exist several quadrac reciprocity laws depending on the quadratic polyonoms, resp. is the quadratic reciprocity law the same for gaussian and eisenstein primes.
With Eisenstein primes, you'd more likely be dealing with cubic reciprocity.

It should be possible to formulate quadratic reciprocity for the Eisenstein integers. I'm not sure how, offhand.

 2019-05-01, 14:33 #4 Nick     Dec 2012 The Netherlands 2·7·113 Posts See chapter 9 "Cubic and Biquadratic Reciprocity" in the famous book "A Classical Introduction to Modern Number Theory" (2nd edition) by Ireland & Rosen (published by Springer). https://www.springer.com/us/book/9780387973296
2019-05-01, 15:53   #5
GP2

Sep 2003

A1716 Posts

Quote:
 Originally Posted by Dr Sardonicus With Eisenstein primes, you'd more likely be dealing with cubic reciprocity. It should be possible to formulate quadratic reciprocity for the Eisenstein integers. I'm not sure how, offhand.
Maybe the "Eisenstein integers" section of the Wikipedia article about "Cubic reciprocity" might be a start.

 2019-05-02, 16:14 #6 Dr Sardonicus     Feb 2017 Nowhere 2×2,081 Posts
 2019-06-03, 23:26 #7 bhelmes     Mar 2016 5·59 Posts A peaceful night for you, is there a special quadratic reciprocity law for the polynomial f(n)=2n^2-1 ? It might be also interesting for other persons. Greetings from the "even primes" Bernhard

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