Bases for cyclotomic units over function fields (Q2726649)

The structure of the cyclotomic units in the number field case is obtained from the universal punctured distribution. In [Compos. Math. 71, 13-27 (1989; Zbl 0687.12003)], \textit{R. Gold} and \textit{J. Kim} found an explicit basis of the universal punctured even distribution and, from it, a basis of the group of cyclotomic units. \textit{S. Galovich} and \textit{M. Rosen} [J. Number Theory 13, 363-375 (1981; Zbl 0473.12014)] introduced the notion of cyclotomic units for cyclotomic function fields. In the paper under review, the authors find a basis of the universal even punctured distribution in function fields and, from it, a basis of the group of cyclotomic units. NEWLINENEWLINENEWLINEThe principal ideas are similar to those in the paper of Gold and Kim. The principal result is that if \(M \in R _T = {\mathbb{F}} _q [T]\), \(\lambda _M\) is a primitive \(M\)-th root of the Carlitz module \(\Lambda _M\) and \(U _M\) is the group of cyclotomic units of \({\mathbb{F}} _q [T](\lambda _M)\), then \(U_M \cong \varphi(G_1) \times {\mathbb{F}} ^\ast _q\) where \(G _1\) is a subgroup of the universal punctured even distribution \((A ^\circ _M)^+\) and \(\varphi\) is a homomorphism of \((A ^\circ _M)^+\) into \(V_M/ {\mathbb{F}}^\ast _q\) where \(V_M\) is the subgroup of \({\mathbb{F}}_q[T](\lambda _M) ^\ast\) generated by \(\{\lambda _M ^B \mid B \in R_T/M R_T\), \(B\not\equiv 0 \bmod M\} \cup {\mathbb{F}}_q ^\ast\). NEWLINENEWLINENEWLINEFinally, it is proved that \(U _{QM}^G = U _M\) where \(G = \text{ Gal} ({\mathbb{F}}_q[T](\Lambda _{QM})/{\mathbb{F}}_q[T](\Lambda _M))\).