×

An extension of the Jacobi symbol due to Gauss. (English) Zbl 1047.11005

Summary: In a posthumous paper of Gauss the definition of the (nowadays called) Jacobi symbol for biquadratic residues in \(\mathbb Q(i)\) is based on a generalisation of the Gauss lemma and at the same time extended to all denominators prime to the numerator. The author shows what kind of characters result from an analogous extension of the Jacobi symbol for \(n\)th power residues in any suitable number field.

MSC:

11A15 Power residues, reciprocity
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barnes, F. W., A permutation reciprocity law, Ars Combin. A, 29, 155-159 (1990) · Zbl 0716.11004
[2] Cartier, P., Sur une généralisation des symboles de Legendre-Jacobi, Enseign. Math., 16, 31-48 (1970) · Zbl 0195.05802
[3] Cartier, P., Sur une généralisation du transfert en théorie des groupes, Enseign. Math., 16, 49-57 (1970) · Zbl 0195.04001
[4] Eisenstein, G., Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus dritten Wurzeln der Einheit zusammengesetzten complexen Zahlen, J. Reine Angew. Math., 27, 289-310 (1844) · ERAM 027.0801cj
[5] Gauss, C. F., Theorematis arithmetici demonstratio nova, Comm. Soc. Reg. Sci. Gottingensis, 16 (1808)
[6] Gauss, C. F., Theoria residuorum biquadraticorum, commentatio secunda, Comm. Soc. Reg. Sci. Gottingensis Rec., 7 (1832)
[7] Gauss, C. F., Zur Theorie der biquadratischen Reste, Fragment, Werke II (1863), p. 313-325
[8] Hasse, H., Number Theory (1980), Springer-Verlag: Springer-Verlag New York
[9] Ireland, K.; Rosen, M., A Classical Introduction to Modern Number Theory (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0482.10001
[10] Kantz, G., Über einen Satz aus der Theorie der biquadratischen Reste, Deutsche Math., 5, 269-272 (1940) · JFM 66.0138.06
[11] Lerch, M., Sur un théorème arithmétique de Zolotarev, Ceska Akad. Prague Bull. Inter. Cl. Math., 3, 34-37 (1896) · JFM 27.0141.02
[12] Leutbecher, A., Das Gauss’sche Lemma ist Beispiel einer Verlagerung, Arch. Math., 23, 151-153 (1972) · Zbl 0251.10003
[13] Reichardt, H., Eine Bemerkung zur Theorie des Jacobischen Symbols, Math. Nachr., 19, 172-175 (1958) · Zbl 0088.25501
[14] Rousseau, G., On the Jacobi symbol, J. Number Theory, 48, 109-111 (1994) · Zbl 0814.11002
[15] Schering, E., Zur Theorie der quadratischen Reste, Acta Math., 1, 153-170 (1882) · JFM 15.0141.01
[16] Watanabe, T., Random walks on \(SL (2, F_2)\) and Jacobi symbols of quadratic residues, Advances in Combinatorial Methods and Applications to Probability and Statistics (1997), Birkhäuser: Birkhäuser Boston, p. 125-134 · Zbl 0887.11021
[17] Waterhouse, W. C., A tiny note on Gauss’s lemma, J. Number Theory, 30, 105-107 (1988) · Zbl 0663.12002
[18] Zolotareff, M., Nouvelle démonstration de la loi de réciprocité de Legendre, Nouv. Ann. Math., 11, 354-362 (1872)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.