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

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors’ Addresses Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login Author Index Coming Soon Contacts ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print)
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