Power Reciprocity for Binomial Cyclotomic Integers

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We give an explicit expression for the inversion factor (α/β)l(β/α)-1lof thelth power residue symbol over the cyclotomic field of l th roots of unity, whenαandβare binomial cyclotomic integersx+yζnrelatively prime to each other and tol. Herelis an odd prime number,ζa primitivelth root of unity andx, y∈Z We note that Eisenstein's reciprocity law extends to the case where primary binomial integers replace rational integers. As an application, we obtain necessary and sufficient congruence conditions for a rational integer to be anlth power residue modulo some prime numbers of the form (xl+1)/(x+1).

Original languageEnglish (US)
Pages (from-to)245-256
Number of pages12
JournalJournal of Number Theory
Issue number2
StatePublished - Aug 1998

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory


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