Genus theory of global fields. (Théorie des genres des corps globaux.) (French) Zbl 0982.11062

The authors present a unified treatment of genus theory for finite extensions of global fields (number fields or function fields of curves over finite fields), which need not be Galois. For function fields \(K\), one first needs to declare a fixed place to be “at infinity”; one can then introduce the analogue of sign functions, which generalizes the notion of leading coefficient of a polynomial. The quotient of \(K^*\) by the subgroup of its totally positive elements (w.r.t. the sign functions) provides the link between ideal classes and narrow ideal classes. The direct sum of this group with the group of ideals is called the group of signed ideals (= divisors); its classes form precisely the restricted class group; one can equally well define all of this for \(S\)-ideal classes, where \(S\) is a fixed set of finite places of \(K\).
The \(S\)-genus field \(G\) of a finite extension \(L\) of \(K\) is the largest subfield of the \(S\)-Hilbert class field of \(L\) that is abelian over \(K\). The authors establish a generalization of a formula of Y. Furuta [Nagoya Math. J. 29, 281-285 (1967; Zbl 0166.05901)] for Galois extensions and L. J. Goldstein [Nagoya Math. J. 45, 119-127 (1972; Zbl 0234.12002)] for abelian extensions that expresses the degree of \(LH\) over \(L\) purely in terms of data associated to the extension \(L/K\) (\(S\)-class number of \(K\), number of \(S\)-units and \(S\)-local norms from \(L/K\) and data associated to the maximal abelian subextension of \(L/K\) and its localizations). Finally, the Galois group of \(G\) over the Hilbert class field of \(K\) can be determined likewise by purely local data. The formulas are made more explicit in case \(K\) is a function field with one place at infinity. Finally, the \(S\)-central class field of \(L/K\) is defined and for Galois extensions, its degree determined by establishing a link with \(H\).


11R37 Class field theory
11R58 Arithmetic theory of algebraic function fields
11R29 Class numbers, class groups, discriminants
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