One of the most important problems in number theory is to give an explicit construction of a given global field $L$. If $L=\mathbb{Q}$, the maximal abelian extension of $\mathbb{Q}$ is the union of all cyclotomic extensions. If $L$ is an imaginary quadratic field, the maximal abelian extension of $L$ can be constructed by adjoining the $j$-values and the Weber function at points of finite order for some elliptic curve with complex multiplication. In the function field case, there are similar results to the above. The purpose of our talk is to construct the maximal abelian extension of a given totally imaginary function field. [<6>]