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**The Hilbert class field in function fields.**
*(English)*
Zbl 0632.12017

Let K be an algebraic function field of one variable with a finite constant field F. The purpose of this paper is to construct a field extension of K which behaves like the Hilbert class field in the number field case and to find substitutes for some classical results related to the Hilbert class field.

Let S be a finite set of prime divisors of K. Let A be the ring of elements in K whose poles are in S. A is a Dedekind domain with finite class group. Let \(K_ s\) be a separable closure of K. If \(L\subset K_ s\) and L:K is finite let B be the integral closure of A in \(L^ s\). The Hilbert class field \(K_ A\) of K with respect to A is the maximal unramified abelian extension of K in \(K_ s\) in which every \(P\in S\) splits completely. The author shows that the Artin symbol induces an isomorphism between the class group of A and \(Gal(K_ A/K)\). The constant field of \(K_ A\) is \(F_ d\) where \(F_ d\) is the unique subfield of \(\bar K\) (the maximal constant field extension of K) of dimension \(d\) over F.

Many applications are discussed: e.g. a substitute for the principal ideal theorem is proved, the analogue of the class field tower problem and Iwasawa theory is formulated, the notion of a totally imaginary extension is introduced, the units of A, the regulator, the zeta-function and its Taylor expansion at zero are considered, an analogue of Leopoldt’s Spiegelungssatz and a version of a theorem of A. Scholz on the 3-rank of the class group of quadratic number fields are shown.

Let S be a finite set of prime divisors of K. Let A be the ring of elements in K whose poles are in S. A is a Dedekind domain with finite class group. Let \(K_ s\) be a separable closure of K. If \(L\subset K_ s\) and L:K is finite let B be the integral closure of A in \(L^ s\). The Hilbert class field \(K_ A\) of K with respect to A is the maximal unramified abelian extension of K in \(K_ s\) in which every \(P\in S\) splits completely. The author shows that the Artin symbol induces an isomorphism between the class group of A and \(Gal(K_ A/K)\). The constant field of \(K_ A\) is \(F_ d\) where \(F_ d\) is the unique subfield of \(\bar K\) (the maximal constant field extension of K) of dimension \(d\) over F.

Many applications are discussed: e.g. a substitute for the principal ideal theorem is proved, the analogue of the class field tower problem and Iwasawa theory is formulated, the notion of a totally imaginary extension is introduced, the units of A, the regulator, the zeta-function and its Taylor expansion at zero are considered, an analogue of Leopoldt’s Spiegelungssatz and a version of a theorem of A. Scholz on the 3-rank of the class group of quadratic number fields are shown.

Reviewer: H.Opolka

### MSC:

11R58 | Arithmetic theory of algebraic function fields |

11R32 | Galois theory |

11R37 | Class field theory |