Abstract
Let F/K be a field of algebraic functions of one variable, K algebraically closed of characteristic p≠0. Let E/K be a cyclic extension of F/K of degree ℓ, a prime not necessarily different from p. Let ξ, ρ denote the ℓ-ranks of the null class groups of E and F respectively. If E/F is ramified, Deuring proved that ξ=ℓρ+(t−δ)(ℓ−1) where t is the number of ramified primes and δ is 1 or 2 according as ℓ equals p or not. Šafarevič proved the same relation in the unramified case for ℓ=p where ξ, ρ denote the ranks of the Hasse-Witt matrices. Subrao gave a unified proof for ℓ=p. Rosen and Sullivan 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.
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References
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Ch.C. Tsen: Divisionsalgebren über Funktionenkörpern, Göttinger Dissertation (1933).
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The author is grateful to the Alexander von Humboldt-Stiftung, Federal Republic of Germany, for supporting this research done at the Mathematisches Institut der Universität, Heidelberg.
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Madan, M.L. On a theorem of M. Deuring and I. R. Šafarevič. Manuscripta Math 23, 91–102 (1977). https://doi.org/10.1007/BF01168587
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DOI: https://doi.org/10.1007/BF01168587