The indices of Stickelberger ideals of function fields
Let \({\mathbb F}_ q\) be the finite field of \(q\) elements and let \(k\) be a congruence function field over \({\mathbb F}_ q\). Fix a place \(\infty\) of degree \(d _ \infty\) and let \({\mathbb F}_\infty\) be the residue field at \(\infty\). Let \({\mathbb A}\) be the Dedekind domain of the regular function away from \(\infty\). For an integral ideal \({\mathcal M}\) of \({\mathbb A}\), let \(K_ {\mathcal M}\) be the cyclotomic function field and \(K_ {\mathcal M}^ +\) its maximal real subfield. In [Math. Z. 239, 425--440 (2002; Zbl 1034.11062)], \textit{L. Yin} gave a new definition of the Stickelberger ideals \(I\) and \(I^ +\) in the group rings \({\mathbb Z}[G]\) and \({\mathbb Z}[G^ +]\) respectively, where \(G= \text{Gal}(K_ {\mathcal M}/k)\) and \(G^ += \text{Gal} (K_ {\mathcal M}^ +/k)\), by using the Stickelberger elements associated to \(K_ {\mathcal M}/k\) and \(K_ {\mathcal M}^ +/k\). In the paper under review the authors generalize the index formula proved by Yin for the case \(d_ \infty=1\) to general \(d_ \infty\). The indices \([{\mathbb Z} [G]:I]\) and \([{\mathbb Z}[G^ +]:I^ +]\) are computed explicitly.
scientific article; zbMATH DE number 1837065