Let k be a global function field with a fixed prime divisor ∞. Let A be the ring of integers outside ∞. D.Hayes generated three Class Fields, Hilbert Class Field $H_A$, Normalizing Field $\indetilde{H}_A$, and $K_m=\indetilde{H}_A(\Lambda_m)$ over k, using elliptic A-module of rank 1 of generic characteristic i and sgn-normalized elliptic A-module([1],[2]).
In this paper, we extend the above result to the case that A is replaced by an order R of A. Similary we generate three Class Fields, Hilbert Class Field of R, $H_R$, Normalizing Field $\indetilde{H}_R$, and $\indetilde{K}_m=\indetilde{H}_R(\Lambda_m)$ over k.
We also identify them by using Class Field Theory as follows,
$H_R$ $\longleftrightarrow$ $J_R$
$\indetilde{H}_R$ $\longleftrightarrow$ $\indetilde{J}_R$
$K_m$ $\longleftrightarrow$ $\indetilde{J}_m$
where $J_R=k^{\times}\cdot\pi^Z_{\infty}\cdotU_R$; $\indetilde{J}_R=k^{\times}\cdot\pi^Z_{\infty}\cdot\indetilde{U}_R$; $\indetilde{J}_m=k^{\times}\cdot\pi^Z_{\infty}\cdot\indetilde{U}_m$.
이 논문에서, 우리는 함수체(函數體)위에서의 세가지의 유체(類體)-힐버트 유체 $H_R$, 정규유체 $\indetilde{H}_R$, 그리고 $K_m$ = $\indetilde{H}_R(\Lambda)-$를 구체적으로 생성 하였다. 그리고 유체론(類體論)을 이용 하여서 위의 세 유체를 다음과 같이 등화(等化) 하였다.
$H_R \longleftrightarrow J_R$
$\indetilde{H}_R \longleftrightarrow \indetilde{J}_R$
$K_m \longleftrightarrow \indetilde{J}_m$
