Title & Authors
Choi, Eun-Mi;

Abstract
If k is a subfield of $\small{\mathbb{Q}(\varepsilon_m)}$ then the cohomology group $\small{H^2(k(\varepsilon_n)/k)}$ is isomorphic to $\small{H^2(k(\varepsilon_{n$ with gcd(m, n') = 1. This enables us to reduce a cyclotomic k-algebra over $\small{k(\varepsilon_n)}$ to the one over $\small{k(\varepsilon_{n$. A radical extension in projective Schur algebra theory is regarded as an analog of cyclotomic extension in Schur algebra theory. We will study a reduction of cohomology group of radical extension and show that a Galois cohomology group of a radical extension is isomorphic to that of a certain subextension of radical extension. We then draw a cohomological characterization of radical group.
Keywords
Language
English
Cited by
References
1.
E. Aljadeff and J. Sonn, Projective Schur algebras have abelian splitting fields, J. Algebra 175 (1995), no. 1, 179-187

2.
E. Aljadeff and J. Sonn, Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras, J. Algebra 220 (1999), no. 2, 401-414

3.
A. Babakhanian, Cohomological methods in group theory, Marcel Dekker Inc., New York, 1972

4.
E. Choi and H. Lee, Crossed product algebras over cyclotomic extension fields, Math. Japon. 50 (1999), no. 2, 207-210

5.
E. Choi and H. Lee, The crossed product theorem for projective Schur algebras, Glasg. Math. J. 43 (2001), no. 1, 135-143

6.
G. Janusz, The Schur group of cyclotomic fields, J. Number Theory 7 (1975), no. 3, 345-352

7.
G. Karpilovsky, Projective representations of finite groups, Monographs and Textbooks in Pure and Applied Mathematics, 94, Marcel Dekker Inc., New York, 1985

8.
G. Karpilovsky, Field Theory, Monographs and Textbooks in Pure and Applied Mathematics, 120, Marcel Dekker Inc., New York, 1988

9.
F. Lorenz and H. Opolka, Einfache Algebren und projektive Darstellungen Äuber ZahlkÄorpern, Math. Z. 162 (1978), no. 2, 175-182

10.
I. Reiner, Maximal orders, London Math. Soc. Monogr. Ser. 5, London, Academic Press, New York, 1975

11.
J. P. Serre, Local Fields, Grad. Text. Math. 67, Springer-Verlag, NewYork, 1979

12.
E. Weiss, Cohomology of groups , Academic press, New York, 1969

13.
T. Yamada, The Schur subgroup of the Brauer group, Lect. Notes Math. 397, Springer- Verlag, NewYork, 1974