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Gauss Problem for Function Fields

✍ Scribed by Gilles Lachaud; Serge Vlăduţ

Publisher: Elsevier Science
Year: 2000
Tongue: English
Weight: 173 KB
Volume: 85
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.2000.2523

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✦ Synopsis

According to a celebrated conjecture of Gauss, there are infinitely many real quadratic fields whose ring of integers is principal. We recall this conjecture in the framework of global fields. If one removes any assumption on the degree, this leads to various related problems for which we give solutions; namely, we prove that there are infinite families of principal rings of algebraic functions in positive characteristic, which are extensions of a given one, and with prescribed Galois, or ramification, properties, at least in some particular cases.

2000 Academic Press

In modern language this question can be reformulated along the following lines: The abovementioned determinants are the discriminants of real quadratic fields. If D>0 is such a discriminant, then we denote by Cl(D) the class group of the corresponding real quadratic field K=Q(-D), and

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