Quadratic function fields with invariant class group

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Abstract

Emil Artin studied quadratic extensions of k(x) where k is a prime field of odd characteristic. He showed that there are only finitely many such extensions in which the ideal class group has exponent two and the infinite prime does not decompose. The main result of this paper is: If K is a quadratic imaginary extension of k(x) of genus G, where k is a finite field of order q, in which the infinite prime of k(x) ramifies, and if the ideal class group has exponent 2, then q = 9, 7, 5, 4, 3, or 2 and G ≤ 1, 1, 2, 2, 4, and 8, respectively. The method of Artin's proof gives G ≤ 13, 9, and 9724 for q = 7, 5, and 3, respectively. If the infinite prime is inert in K, both the methods of this paper and Artin's methods give bounds on the genus that are roughly double those in the ramified case.

Original languageEnglish (US)
Pages (from-to)218-228
Number of pages11
JournalJournal of Number Theory
Volume9
Issue number2
DOIs
StatePublished - May 1977
Externally publishedYes

ASJC Scopus subject areas

  • Algebra and Number Theory

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