Central Extensions and Hasse Norm Principle over Function Fields
Sunghan Bae, Hwanyup Jung · Tokyo Journal of Mathematics · 2001
We say that Hasse norm principle holds for $K/k$ if $ k^{*}\cap$ $N_{K/k}J(K)=N_{K/k}K^{*}$ .In number field case, several authors have studied the validity of Hasse norm principle for abelian extensions.It is very closely tied up with central extensions.In [Ge2], Gerth gave necessary and sufficient conditions for Hasse norm principle to hold for cyclotomic fields.In [K], Kagawa gave conditions for Hasse norm principle to hold for maximal real subfields of cyclotomic fields.Central extensions are also useful in studying ideal class groups $([CoRo]$ , [Fr], [Fu3]).Let $k=F_{q}(T)$ be the rational function field over finite field $F_{q}$ , where $q=p^{f},$ $p=$ $char(k)$ and $A=F_{q}[T]$ .For any monic polynomial $m\in A$ , let $k(\Lambda_{m})$ be the m-th cyclotomic function field and $k(\Lambda_{m})^{+}$ its maximal real subfield.In this paper, we define central class fields of Galois extensions of function fields, give necessary and sufficient conditions for Hasse norm principle to hold for $k(\Lambda_{m})$ and $k(\Lambda_{m})^{+}$ , and find lower bounds for the $\ell$ -rank of ideal class groups of $k(\Lambda_{m})$ and $k(\Lambda_{m})^{+}$ .1. Central class fleld and Genus field.Let $k$ be a global function field over a finite field $F_{q}$ .Let $\infty$ be a place of degree 1 of $k$ and $\mathcal{O}_{k}$ the ring of regular elements outside $\infty$ of $k$ .Let $E_{k}$ be the unit group of $\mathcal{O}_{k}$ , which is just $F_{q}^{*}$ .We write $k_{\infty}$ to be the completion of $k$ at $\infty$ .We fix a sing function $sgn$ : $k_{\infty}^{*}\rightarrow F_{q}^{*}$ and choose a uniformizer $\pi$ of $k_{\infty}$ with $sgn(\pi)=1$ .Denote by $\tilde{C}$ the field $k_{\infty}(q-\sqrt[1]{-\pi})$ .In the following we mean by an extension of $k$ , a separable extension of $k$ for which any embeddings into $k_{\infty}^{ac}$ lies in $\tilde{C}$ viewing as a subfield of $k_{\infty}^{ac}$ .