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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A theorem on biquadratic reciprocity


Author: Ezra Brown
Journal: Proc. Amer. Math. Soc. 30 (1971), 220-222
MSC: Primary 10.68
DOI: https://doi.org/10.1090/S0002-9939-1971-0280462-5
MathSciNet review: 0280462
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Abstract: The following theorem on biquadratic reciprocity is proved: if $p \equiv q \equiv 1 \pmod 4$ are primes for which $(p|q) = 1$, and if $p = {r^2} + q{s^2}$ for some integers r and s, then \[ \begin {array}{*{20}{c}} \hfill {{{(p|q)}_4}{{(q|p)}_4} = 1,\qquad {\text {if}}\;q \equiv 1\;\pmod 8;} \\ \hfill { = {{( - 1)}^s},\quad {\text {if}}\;q \equiv 5 \pmod 8.} \\ \end {array} \] Simple expressions for the biquadratic character of some small primes are also obtained.


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Keywords: Power residues, biquadratic residues, reciprocity, quadratic diophantine equations
Article copyright: © Copyright 1971 American Mathematical Society