AN ACTION OF A GALOIS GROUP ON A TENSOR PRODUCT

Title & Authors
AN ACTION OF A GALOIS GROUP ON A TENSOR PRODUCT
Hwang, Yoon-Sung;

Abstract
Let K be a Galois extension of a field F with G = Gal(K/F). Let L be an extension of F such that $\small{K\;{\otimes}_F\;L\;=\; N_1\;{\oplus}N_2\;{\oplus}{\cdots}{\oplus}N_k}$ with corresponding primitive idempotents $\small{e_1,\;e_2,{\cdots},e_k}$, where Ni's are fields. Then G acts on $\small{\{e_1,\;e_2,{\cdots},e_k\}}$ transitively and $\small{Gal(N_1/K)\;{\cong}\;\{\sigma\;{\in}\;G\;/\;{\sigma}(e_1)\;=\;e_1\}}$. And, let R be a commutative F-algebra, and let P be a prime ideal of R. Let T = $\small{K\;{\otimes}_F\;R}$, and suppose there are only finitely many prime ideals $\small{Q_1,\;Q_2,{\cdots},Q_k}$ of T with $\small{Q_i\;{\cap}\;R\;=\;P}$. Then G acts transitively on $\small{\{Q_1,\;Q_2,{\cdots},Q_k\},\;and\;Gal(qf(T/Q_1)/qf(R/P))\;{\cong}\;\{\sigma{\in}\;G/\;{\sigma}-(Q_1)\;=\;Q_1\}}$ where qf($\small{T/Q_1}$) is the quotient field of $\small{T/Q_1}$.⠌㔀؀㘴ㄮ㔻᠀䡯浥‭⁦慭楬礠浡湡来浥湴
Keywords
Galois extension;Tensor product;
Language
English
Cited by
References
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A. Frohlich and M. J. Taylor, Algebraic Number Theory, Cambridge Univ. Press, Cambridge, 1991

2.
T. W. Hungerford, Algebra, Springer Verlag, New York, 1980