Imaginary bicyclic biquadratic function fields in characteristic two
type: Item
instance of: scholarly article
title: Imaginary bicyclic biquadratic function fields in characteristic two
zbMATH Open document ID: 0961.11041
author: Yves Aubry
author: Dominique Le Brigand
published in: Journal of Number Theory
publication date: 1999-09-23T00:00:00Z
review text: The authors prove an analog of a result proved in the number field case by \textit{E. Brown} and \textit{C. J. Parry} [J. Reine Angew. Math. 266, 118-120 (1974; Zbl 0287.12015)] and in the odd characteristic function field case by \textit{X. Zhang} [Sci. Sin., Ser. A 31, No. 8, 908-915 (1988; Zbl 0664.12013)]. The objective of this article is to classify all imaginary bicyclic biquadratic extensions, \(L\), of the fields \(k=F(x)\) of class number one where \(F\) is a finite field of characteristic two. A class number formula essentially identical to that of the number field case is obtained. It follows from this formula that at most one of the three quadratic subfields of \(L\) can have class number two, while the others must have class number one. The authors then make use of known descriptions of quadratic function fields of class numbers one and two to obtain their results. It is interesting to note that there are infinitely many biquadratic fields, \(L\), of class number one, but only finitely many for a fixed field of constants, \(F\). If the genus of \(L\) is zero then the list is finite and the field of constants has either 2 or 4 elements. All biquadratic fields of class number one have genus 5 or less and contain a quadratic subfield of genus zero.
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1309265
zbMATH Keywords: imaginary bicyclic biquadratic extensions
zbMATH Keywords: class number
zbMATH Keywords: class number two
zbMATH Keywords: class number one
zbMATH Keywords: function fields
MaRDI profile type: Publication
OpenAlex ID: W2076815454
cites work: Q4718153
cites work: The imaginary bicyclic biquadratic fields with class-number 1.
cites work: Q4336972
cites work: Algebraic function fields with small class number
cites work: Algebraic function fields of class number one
cites work: On unique factorization in certain rings of algebraic functions
cites work: The Hilbert class field in function fields
cites work: Analytische Zahlentheorie in Körpern der Charakteristik \(p\)
cites work: Fibre products of Artin-Schreier curves and generalized Hamming weights of codes
cites work: Q3815394
cites work: Q3815393
cites work: Q3784159
DOI: 10.1006/JNTH.1999.2377
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description: scientific article; zbMATH DE number 1309265