A theorem on biquadratic reciprocity
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- by Ezra Brown
- Proc. Amer. Math. Soc. 30 (1971), 220-222
- DOI: https://doi.org/10.1090/S0002-9939-1971-0280462-5
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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.References
- Ezra Brown, Representations of discriminantal divisors by binary quadratic forms, J. Number Theory 3 (1971), 213–225. MR 285485, DOI 10.1016/0022-314X(71)90039-4
- Klaus Burde, Ein rationales biquadratisches Reziprozitätsgesetz, J. Reine Angew. Math. 235 (1969), 175–184 (German). MR 241354, DOI 10.1515/crll.1969.235.175
Bibliographic Information
- © Copyright 1971 American Mathematical Society
- 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