A generalization of Scholz’s reciprocity law

Mark Budden[1]; Jeremiah Eisenmenger[2]; Jonathan Kish[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 (2007)

  • Volume: 19, Issue: 3, page 583-594
  • ISSN: 1246-7405

Abstract

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We provide a generalization of Scholz’s reciprocity law using the subfields and of , of degrees and over , respectively. The proof requires a particular choice of primitive element for over and is based upon the splitting of the cyclotomic polynomial over the subfields.

How to cite

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Budden, Mark, Eisenmenger, Jeremiah, and Kish, Jonathan. "A generalization of Scholz’s reciprocity law." Journal de Théorie des Nombres de Bordeaux 19.3 (2007): 583-594. <http://eudml.org/doc/249921>.

@article{Budden2007,
abstract = {We provide a generalization of Scholz’s reciprocity law using the subfields $K_\{2^\{t-1\}\}$ and $K_\{2^t\}$ of $\mathbb\{Q\}(\zeta _p )$, of degrees $2^\{t-1\}$ and $2^\{t\}$ over $\mathbb\{Q\}$, 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 $\Phi _p (x)$ over the subfields.},
affiliation = {Department of Mathematics Armstrong Atlantic State University 11935 Abercorn St. Savannah, GA USA 31419; Department of Mathematics University of Florida PO Box 118105 Gainesville, FL USA 32611-8105; Department of Mathematics University of Colorado at Boulder Boulder, CO USA 80309},
author = {Budden, Mark, Eisenmenger, Jeremiah, Kish, Jonathan},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Scholz' reciprocity law; rational reciprocity laws; cyclotomic number fields},
language = {eng},
number = {3},
pages = {583-594},
publisher = {Université Bordeaux 1},
title = {A generalization of Scholz’s reciprocity law},
url = {http://eudml.org/doc/249921},
volume = {19},
year = {2007},
}

TY - JOUR
AU - Budden, Mark
AU - Eisenmenger, Jeremiah
AU - Kish, Jonathan
TI - A generalization of Scholz’s reciprocity law
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2007
PB - Université Bordeaux 1
VL - 19
IS - 3
SP - 583
EP - 594
AB - We provide a generalization of Scholz’s reciprocity law using the subfields $K_{2^{t-1}}$ and $K_{2^t}$ of $\mathbb{Q}(\zeta _p )$, of degrees $2^{t-1}$ and $2^{t}$ over $\mathbb{Q}$, 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 $\Phi _p (x)$ over the subfields.
LA - eng
KW - Scholz' reciprocity law; rational reciprocity laws; cyclotomic number fields
UR - http://eudml.org/doc/249921
ER -

References

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