#### Vol. 3, No. 3, 2010

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Rational residuacity of primes

### Mark Budden, Alex Collins, Kristin Ellis Lea and Stephen Savioli

Vol. 3 (2010), No. 3, 249–257
##### Abstract

The most natural extensions to the law of quadratic reciprocity are the rational reciprocity laws, described using the rational residue symbol. In this article, we provide a reciprocity law from which many of the known rational reciprocity laws may be recovered by picking appropriate primitive elements for subfields of $ℚ\left({\zeta }_{p}\right)$. As an example, a new generalization of Burde’s law is provided.

##### Keywords
reciprocity laws, ramification of prime ideals, cyclotomic fields
##### Mathematical Subject Classification 2000
Primary: 11A15
Secondary: 11R32, 11R18