A generalization of Scholz’s reciprocity law
Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 3, pp. 583-594.

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 (ζ p ) de degrés 2 t-1 et 2 t sur , 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.

We provide a generalization of Scholz’s reciprocity law using the subfields K 2 t-1 and K 2 t of (ζ p ), of degrees 2 t-1 and 2 t over , 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.

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

[1] D. Buell and K. Williams, Is There an Octic Reciprocity Law of Scholz Type?. Amer. Math. Monthly 85 (1978), 483–484. | MR | Zbl

[2] D. Buell and K. Williams, An Octic Reciprocity Law of Scholz Type. Proc. Amer. Math. Soc. 77 (1979), 315–318. | MR | Zbl

[3] D. Estes and G. Pall, Spinor Genera of Binary Quadratic Forms. J. Number Theory 5 (1973), 421–432. | MR | Zbl

[4] R. Evans, Residuacity of Primes. Rocky Mountain J. of Math. 19 (1989), 1069–1081. | MR | Zbl

[5] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory. 2 nd edition, Graduate Texts in Mathematics 84, Springer-Verlag, 1990. | MR | Zbl

[6] G. Janusz, Algebraic Number Fields. 2 nd ed., Graduate Studies in Mathematics 7, American Mathematical Society, Providence, RI, 1996. | MR | Zbl

[7] E. Lehmer, On the Quadratic Character of some Quadratic Surds. J. Reine Angew. Math. 250 (1971), 42–48. | MR | Zbl

[8] E. Lehmer, Generalizations of Gauss’ Lemma. Number Theory and Algebra, Academic Press, New York, 1977, 187–194. | Zbl

[9] E. Lehmer, Rational Reciprocity Laws. Amer. Math. Monthly 85 (1978), 467–472. | MR | Zbl

[10] F. Lemmermeyer, Rational Quartic Reciprocity. Acta Arith. 67 (1994), 387–390. | MR | Zbl

[11] F. Lemmermeyer, Reciprocity Laws. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. | MR | Zbl

[12] A. Scholz, Über die Lösbarkeit der Gleichung t 2 -Du 2 =-4. Math. Z. 39 (1934), 95–111.

[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. | Zbl

[14] K. Williams, On Scholz’s Reciprocity Law. Proc. Amer. Math. Soc. 64 No. 1 (1977), 45–46. | Zbl

[15] K. Williams, K. Hardy, and C. Friesen, On the Evaluation of the Legendre Symbol A+Bm p. Acta Arith. 45 (1985), 255–272. | MR | Zbl

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