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 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.

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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  1. D. Buell and K. Williams, Is There an Octic Reciprocity Law of Scholz Type?. Amer. Math. Monthly 85 (1978), 483–484. Zbl0383.10004MR545588
  2. D. Buell and K. Williams, An Octic Reciprocity Law of Scholz Type. Proc. Amer. Math. Soc. 77 (1979), 315–318. Zbl0417.10002MR545588
  3. D. Estes and G. Pall, Spinor Genera of Binary Quadratic Forms. J. Number Theory 5 (1973), 421–432. Zbl0268.10010MR332653
  4. R. Evans, Residuacity of Primes. Rocky Mountain J. of Math. 19 (1989), 1069–1081. Zbl0699.10012MR1039544
  5. K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory. 2 n d edition, Graduate Texts in Mathematics 84, Springer-Verlag, 1990. Zbl0712.11001MR1070716
  6. G. Janusz, Algebraic Number Fields. 2 n d ed., Graduate Studies in Mathematics 7, American Mathematical Society, Providence, RI, 1996. Zbl0854.11001MR1362545
  7. E. Lehmer, On the Quadratic Character of some Quadratic Surds. J. Reine Angew. Math. 250 (1971), 42–48. Zbl0222.12007MR286777
  8. E. Lehmer, Generalizations of Gauss’ Lemma. Number Theory and Algebra, Academic Press, New York, 1977, 187–194. Zbl0382.10003
  9. E. Lehmer, Rational Reciprocity Laws. Amer. Math. Monthly 85 (1978), 467–472. Zbl0383.10003MR498352
  10. F. Lemmermeyer, Rational Quartic Reciprocity. Acta Arith. 67 (1994), 387–390. Zbl0833.11049MR1301826
  11. F. Lemmermeyer, Reciprocity Laws. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. Zbl0949.11002MR1761696
  12. A. Scholz, Über die Lösbarkeit der Gleichung t 2 - D u 2 = - 4 . Math. Z. 39 (1934), 95–111. Zbl0009.29402
  13. T. Schönemann, Theorie der Symmetrischen Functionen der Wurzeln einer Gleichung. Allgemeine Sätze über Congruenzen nebst einigen Anwendungen derselben. J. Reine Angew. Math. 19 (1839), 289–308. 
  14. K. Williams, On Scholz’s Reciprocity Law. Proc. Amer. Math. Soc. 64 No. 1 (1977), 45–46. Zbl0372.12004
  15. K. Williams, K. Hardy, and C. Friesen, On the Evaluation of the Legendre Symbol A + B m p . Acta Arith. 45 (1985), 255–272. Zbl0524.10002MR808025

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