Imaginary bicyclic biquadratic function fields in characteristic two
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.
imaginary bicyclic biquadratic extensions
scientific article; zbMATH DE number 1309265
scientific article