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I wanna know if a finite Galois extension of a semi-local commutative ring is also semilocal.

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    Assuming that finite Galois extensions are finite ring extensions, if the semilocal ring $A$ can be viewed as a finite extension of a field $k$, then the answer is yes, since finiteness of $B/A$ and $A/k$ implies finiteness of $B/k,$ which guarantees $B$ is semilocal.2012-09-28

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