My question in the line 5 of the proof of $(i)\Rightarrow (ii)$. I wonder why $A^L$ is a split semi-simple $L$-algebra...
Thank you very much!!
Why $A^L$ is a split semi-simple $L$-algebra?
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abstract-algebra
field-theory
representation-theory
finite-fields
extension-field
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1I think they are using the fact that if $A$ is split semisimple over some field $K$, then $A^M$ will be split semisimple over any extension field $M$ of $K$. So $A^L$ is split semisimple, because $A^E$ is and $A^L=(A^E)^L$. – 2017-01-03
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0Why $A^L$ is split semisimple, because $A^E$ is and $A^L=(A^E)^{L}$？ – 2017-01-04
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0My question is just why $A^L$ is split semisimple? – 2017-01-04
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1Is that not because $M_n(K)\otimes_K M\simeq M_n(M)$ for all field extensions $M/K$? If $A^E\simeq\prod_{i=1}^tM_{n_i}(E)$, then $A^L\simeq\ldots$ – 2017-01-04
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0Thank you! What really confused me is $M_n(K)\otimes_KM≃M_n(M)$, it may be Intuitively, but I failed to prove it. – 2017-01-04