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Power Reciprocity for Binomial Cyclotomic Integers

โœ Scribed by Charles Helou

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
260 KB
Volume
71
Category
Article
ISSN
0022-314X

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โœฆ Synopsis

We give an explicit expression for the inversion factor (:ร‚;) l (;ร‚:) &1 l of the l th power residue symbol over the cyclotomic field of lth roots of unity, when : and ; are binomial cyclotomic integers x+ y`n relatively prime to each other and to l.

Here l is an odd prime number, `a primitive lth root of unity and x, 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 an l th power residue modulo some prime numbers of the form (x l +1)ร‚(x+1).

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