Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2007; 44(1): 151-167

Printed January 1, 2007

Copyright © The Korean Mathematical Society.

Cohomology groups of radical extensions

Eunmi Choi

HanNam University

Abstract

If $k$ is a subfield of $\mathbb Q(\varepsilon_m)$ then the cohomology group $H^2(k$ $(\varepsilon_n)/k)$ is isomorphic to $H^2(k(\varepsilon_{n^\prime})/k)$ with $\gcd(m,n^\prime)=1$. This enables us to reduce a cyclotomic $k$-algebra over $k(\varepsilon_n)$ to the one over $k(\varepsilon_{n^\prime})$. 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: cohomology group, radical extension

MSC numbers: Primary 16K50, 20G10

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