On a theorem of M. Deuring and I. R. Šafarevič
Let \(F/K\) be a field of algebraic functions of one variable, \(K\) algebraically closed of characteristic \(p\neq 0\). Let \(E/K\) be a cyclic extension of \(F/K\) of degree \(\ell\), a prime not necessarily different from \(p\). Let \(\xi\), \(\rho\) denote the \(\ell\)-ranks of the null class groups of \(E\) and \(F\) respectively. If \(E/F\) is ramified, \textit{M. Deuring} [Math. Ann. 113, 208--215 (1936; Zbl 0014.29301)] proved that \(\xi=\ell\rho+(t-\delta)(\ell-1)\) where \(t\) is the number of ramified primes and \(\delta\) is 1 or 2 according as \(\ell\) equals \(p\) or not. \textit{I. R. Shafarevich} [Am. Math. Soc., Transl., II. Ser. 4, 59--72 (1956); translation from Mat. Sb., N. Ser. 20(62), 351--363 (1947; Zbl 0041.17101)] proved the same relation in the unramified case for \(\ell=p\) where \(\xi\), \(\rho\) denote the ranks of the Hasse-Witt matrices. \textit{D. Subrao} [Manuscr. Math. 16, 169--193 (1975; Zbl 0321.14017)] gave a unified proof for \(\ell=p\). \textit{M. Rosen} [J. Number Theory 9, 160--174 (1977; Zbl 0371.12013)] and \textit{F. J. Sullivan} [Arch. Math. 26, 253--261 (1975; Zbl 0331.14016)] have proved the theorem in special cases. In this paper, using Galois cohomology, Deuring's method of proof is modified so as to give a proof of the theorem in all cases.
field of algebraic functions of one variable
scientific article; zbMATH DE number 3575702