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 (ζp). 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
Milestones
Received: 27 April 2009
Accepted: 20 September 2010
Published: 16 October 2010

Proposed: Filip Saidak
Communicated by Filip Saidak
Authors
Mark Budden
Department of Mathematics and Computer Science
Western Carolina University
Cullowhee, NC 28723
United States
Alex Collins
Department of Mathematics
Armstrong Atlantic State University
11935 Abercorn St.
Savannah, GA 31419
United States
Kristin Ellis Lea
Department of Mathematics
Armstrong Atlantic State University
11935 Abercorn St.
Savannah, GA 31419
United States
Stephen Savioli
Department of Mathematics
Armstrong Atlantic State University
11935 Abercorn St.
Savannah, GA 31419
United States