A generalization of Scholz’s reciprocity law

Budden, Mark; Eisenmenger, Jeremiah; Kish, Jonathan

doi:10.5802/jtnb.604

Budden, Mark ¹; Eisenmenger, Jeremiah ²; Kish, Jonathan ³

¹ Department of Mathematics Armstrong Atlantic State University 11935 Abercorn St. Savannah, GA USA 31419
² Department of Mathematics University of Florida PO Box 118105 Gainesville, FL USA 32611-8105
³ Department of Mathematics University of Colorado at Boulder Boulder, CO USA 80309

Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 583-594.

Abstract
Résumé

We provide a generalization of Scholz’s reciprocity law using the subfields $K_{2^{t - 1}}$ and $K_{2^{t}}$ of $ℚ (ζ_{p})$ , of degrees $2^{t - 1}$ and $2^{t}$ over $ℚ$ , respectively. The proof requires a particular choice of primitive element for $K_{2^{t}}$ over $K_{2^{t - 1}}$ and is based upon the splitting of the cyclotomic polynomial $Φ_{p} (x)$ over the subfields.

Nous donnons une généralisation de la loi de réciprocité de Scholz fondée sur les sous-corps $K_{2^{t - 1}}$ et $K_{2^{t}}$ de $ℚ (ζ_{p})$ de degrés $2^{t - 1}$ et $2^{t}$ sur $ℚ$ , respectivement. La démonstration utilise un choix particulier d’élément primitif pour $K_{2^{t}}$ sur $K_{2^{t - 1}}$ et est basée sur la division du polynôme cyclotomique $Φ_{p} (x)$ sur les sous-corps.

MR Zbl

DOI: 10.5802/jtnb.604

Author's affiliations:

Budden, Mark ¹; Eisenmenger, Jeremiah ²; Kish, Jonathan ³

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Budden, Mark; Eisenmenger, Jeremiah; Kish, Jonathan. A generalization of Scholz’s reciprocity law. Journal de théorie des nombres de Bordeaux, Volume 19 (2007) no. 3, pp. 583-594. doi : 10.5802/jtnb.604. http://archive.numdam.org/articles/10.5802/jtnb.604/

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