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Old 2019-05-01, 10:12   #1
bhelmes
 
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Default 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
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Old 2019-05-01, 11:06   #2
GP2
 
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Why not link as well:

https://en.wikipedia.org/wiki/Quadratic_reciprocity
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Old 2019-05-01, 13:59   #3
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Quote:
Originally Posted by bhelmes View Post
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.
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Old 2019-05-01, 14:33   #4
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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
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Old 2019-05-01, 15:53   #5
GP2
 
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Quote:
Originally Posted by Dr Sardonicus View Post
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.
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Old 2019-05-02, 16:14   #6
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Quadratic Reciprocity in Number Fields
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Old 2019-06-03, 23:26   #7
bhelmes
 
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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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