## Cyclic Stickelberger cohomology and descent of Kummer extensions

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- by Lindsay N. Childs
- Proc. Amer. Math. Soc.
**90**(1984), 505-510 - DOI: https://doi.org/10.1090/S0002-9939-1984-0733396-2
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## Abstract:

Let $R$ be a field, $S = R[{\rm {\zeta }}]$, ${\rm {\zeta }}$ an $n$th root of unit, $\Delta = {\rm {Gal(}}S/R)$. The group of cyclic Kummer extensions of $S$ on which $\Delta$ acts, modulo those which descend to $R$, is isomorphic to a group of roots of unity and to a second group cohomology group of $\Delta$ whose definition involves a "Stickelberger element".## References

- S. U. Chase, D. K. Harrison, and Alex Rosenberg,
*Galois theory and Galois cohomology of commutative rings*, Mem. Amer. Math. Soc.**52**(1965), 15–33. MR**195922** - S. U. Chase and Alex Rosenberg,
*A theorem of Harrison, Kummer theory, and Galois algebras*, Nagoya Math. J.**27**(1966), 663–685. MR**210751** - L. N. Childs,
*On normal Azumaya algebras and the Teichmuller cocycle map*, J. Algebra**23**(1972), 1–17. MR**311701**, DOI 10.1016/0021-8693(72)90042-7
S. Eilenberg and S. Mac Lane, - A. Fröhlich,
*Stickelberger without Gauss sums*, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975) Academic Press, London, 1977, pp. 589–607. MR**0450227** - D. K. Harrison,
*Abelian extensions of commutative rings*, Mem. Amer. Math. Soc.**52**(1965), 1–14. MR**195921** - Paulo Ribenboim,
*13 lectures on Fermat’s last theorem*, Springer-Verlag, New York-Heidelberg, 1979. MR**551363**
S. Chase, - Henri Cartan and Samuel Eilenberg,
*Homological algebra*, Princeton University Press, Princeton, N. J., 1956. MR**0077480**

*Normality of algebras and the Teichmüller cocycle map*, Trans. Amer. Math. Soc.

**64**(1948), 1-20.

*Galois objects and extensions of Hopf algebras*, Lecture Notes in Math., vol. 97, Springer-Verlag, New York, 1969.

## Bibliographic Information

- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**90**(1984), 505-510 - MSC: Primary 12F10; Secondary 13B05
- DOI: https://doi.org/10.1090/S0002-9939-1984-0733396-2
- MathSciNet review: 733396