Journal de Th´
eorie des Nombres
de Bordeaux
19
(2007), 583–594
A generalization of Scholz’s reciprocity law
par
Mark BUDDEN, Jeremiah EISENMENGER
et
Jonathan
KISH
R´esum´e. Nous donnons une g´en´eralisation de la loi de r´eciprocit´e de Scholz fond´ee sur les sous-corpsK2t−1etK2tdeQ(ζp) de degr´es
2t−1 et 2t sur Q, respectivement. La d´emonstration utilise un
choix particulier d’´el´ement primitif pourK2tsurK
2t−1et est bas´ee
sur la division du polynˆome cyclotomique Φp(x) sur les sous-corps.
Abstract. We provide a generalization of Scholz’s reciprocity law using the subfieldsK2t−1 and K2t of Q(ζp), of degrees 2t−1
and 2toverQ, respectively. The proof requires a particular choice
of primitive element for K2t over K
2t−1 and is based upon the
splitting of the cyclotomic polynomial Φp(x) over the subfields.
MarkBudden
Department of Mathematics Armstrong Atlantic State University 11935 Abercorn St.
Savannah, GA USA 31419
E-mail:[email protected]
URL:http://www.math.armstrong.edu/faculty/budden
JeremiahEisenmenger
Department of Mathematics University of Florida PO Box 118105
Gainesville, FL USA 32611-8105
E-mail:[email protected]
JonathanKish
Department of Mathematics University of Colorado at Boulder Boulder, CO USA 80309
E-mail:[email protected]
