On the index of a number field

Nart, Enric

doi:10.1090/S0002-9947-1985-0779058-2

On the index of a number field
HTML articles powered by AMS MathViewer

by Enric Nart

Trans. Amer. Math. Soc. 289 (1985), 171-183

DOI: https://doi.org/10.1090/S0002-9947-1985-0779058-2

PDF | Request permission

Abstract:

Arithmetic invariants are found which determine the index $i(K)$ of a number field $K$. They are used to obtain an explicit formula under certain restrictions on $K$. They provide also a complete explanation of a phenomenon conjectured by Ore [${\mathbf {8}}$] and showed by Engstrom in a particular case [${\mathbf {2}}$].

References

Über den Zusammenhang der Theorie der Ideale und der Theorie der höhere Kongruenzen

23

H. T. Engstrom, On the common index divisors of an algebraic field, Trans. Amer. Math. Soc. 32 (1930), no. 2, 223–237. MR 1501535, DOI 10.1090/S0002-9947-1930-1501535-0
Helmut Hasse, Number theory, Akademie-Verlag, Berlin, 1979. Translated from the third German edition of 1969 by Horst Günter Zimmer. MR 544018

Über den grössten gemeinsamen Teiler aller Zahlen welche durch eine ganze Funktion von n Veränderlichen darstellbar sind

116

Marc Krasner, Nombre des extensions d’un degré donné d’un corps ${\mathfrak {p}}$-adique, Les Tendances Géom. en Algèbre et Théorie des Nombres, Éditions du Centre National de la Recherche Scientifique (CNRS), Paris, 1966, pp. 143–169 (French). MR 0225756
Władysław Narkiewicz, Elementary and analytic theory of algebraic numbers, Monografie Matematyczne, Tom 57, PWN—Polish Scientific Publishers, Warsaw, 1974. MR 0347767
Öystein Ore, Zur Theorie der Algebraischen Körper, Acta Math. 44 (1923), no. 1, 219–314 (German). MR 1555187, DOI 10.1007/BF02403925
Öystein Ore, Newtonsche Polygone in der Theorie der algebraischen Körper, Math. Ann. 99 (1928), no. 1, 84–117 (German). MR 1512440, DOI 10.1007/BF01459087
Jan Śliwa, On the nonessential discriminant divisor of an algebraic number field, Acta Arith. 42 (1982/83), no. 1, 57–72. MR 678997, DOI 10.4064/aa-42-1-57-72
A. A. Sukallo, On determination of the index of a field of algebraic numbers, Rostov. Gos. Univ. Uč. Zap. Fiz.-Mat. Fak. 32 (1955), no. 4, 37–42 (Russian). MR 0075235, DOI 10.1093/restud/32.4.i