Stickelberger’s congruences for absolute norms of relative discriminants. (English) Zbl 1239.11124

Gras, Georges

J. Théor. Nombres Bordx. 22, No. 2, 397-402 (2010).

Let \(L/K\) be a finite extension of number fields, unramified at \(2\), \({\mathfrak d}_{L/K}\) the discriminant, \(c\) the number of complex places of \(L\) which lie above a real place of \(K\), \(\overline{N}_{K/{\mathbb Q}}({\mathfrak d}_{L/K})\) the absolute norm of \({\mathfrak d}_{L/K}\), \(k\) the maximal subfield of \( {\mathbb Q}(\mu_{2^\infty})\) contained in \(K\), \([k:{\mathbb Q}]=2^m,m\geq 0\). The author proves \[ (-1)^c\overline{N}_{K/{\mathbb Q}}({\mathfrak d}_{L/K})\equiv 1\pmod {4\cdot 2^m}. \]

Reviewer: Florin Nicolae (Berlin)

MSC:

11R29	Class numbers, class groups, discriminants
11R37	Class field theory

Keywords:

number fields; discriminants; Stickelberger congruences; class field theory

References:

[1]	G. Gras, Class Field Theory: from theory to practice. SMM, Springer-Verlag, 2003; second corrected printing: 2005. · Zbl 1019.11032
[2]	J. Martinet, Les discriminants quadratiques et la congruence de Stickelberger. Sém. Théorie des Nombres, Bordeaux 1 (1989), 197-204. · Zbl 0731.11061
[3]	S. Pisolkar, Absolute norms of \(p\)-primary units. Jour. de Théorie des Nombres de Bordeaux 21 (2009), 733-740. · Zbl 1214.11131