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Let $K_1, K_2$ be finite field extensions of a field $k$.

Question: Is it true that $A=K_1 \otimes_k K_2$ is isomorphic to a product of group algebras over fields?

Question 2: In case the answer is negative, we still have that $A$ is a symmetric Hopf algebra. What is the group of group-like elements?

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  • $\begingroup$ Previous question with same title: Tensor products of fields $\endgroup$
    – YCor
    Feb 12, 2020 at 10:48
  • $\begingroup$ I am not sure I understand the question. Take $K_2=k$, do you really believe that every finite field extension is a product of group algebras? $\endgroup$
    – abx
    Feb 15, 2020 at 8:49
  • $\begingroup$ @abx In that case $A=K_1 G$ for $G$ the trivial group, or? I think it always works in case $A$ is semi-simple. $\endgroup$
    – Mare
    Feb 15, 2020 at 8:54
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    $\begingroup$ Oh, I see, you allow a group algebra over a field extension. Sorry I missed that. $\endgroup$
    – abx
    Feb 15, 2020 at 13:36

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