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Let $L/K$ be a finite Galois extension with Galois group $G$. For every 2-cocycle $\gamma$ of $G$ with values in $L^\times$ there is the crossed-product $K$-algebra $$S(L,G,\gamma) = \bigoplus_{g\in G} L e_g$$ with the multiplication $(\mu e_g) \cdot (\lambda e_h) = \mu g(\lambda)\gamma(g,h) e_{gh}$ for all $\lambda, \mu \in L$. The $K$-algebra $S(L,G,\gamma)$ is a central simple $K$-algebra. In fact, the isomorphism class of $S(L,G,\gamma)$ only depends on the cohomology class $[\gamma] \in H^2(G, L^\times)$.

Question 1: Is there a cohomological statement about $[\gamma]$ which is equivalent to $S(L,G,\gamma)$ being a division algebra?

Let me try to put this question in a more precise form. There is an obvious necessary condition for $S(L,G,\gamma)$ to be a division algebra.
(C) For every non-trivial subgroup $H \leq G$ the restriction class $[\gamma_{|H}]\in H^2(H,L^\times)$ is non-zero.
(The reason is that $S(L,H, \gamma_{|H})$ is a subring of $S(L,G, \gamma)$ and it has zero-divisors, if $[\gamma_{|H}]=0$. Hence, if $S(L,G,\gamma)$ is a division algebra, then (C) has to hold.).

Question 2: Under which conditions is the necessary condition (C) also sufficient?

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    $\begingroup$ I presume you have checked Chapters 14ff in Pierce's book on Associative Algebras? $\endgroup$
    – user1688
    Commented Feb 11, 2016 at 12:21
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    $\begingroup$ A sufficient condition is that the cocycle has order $n$, where $n$ is the order of $G$ (and the degree of the corresponding algebra). This condition is also necessary over local and global fields or if $n$ is squarefree, but not in general. $\endgroup$
    – Aurel
    Commented Feb 23, 2016 at 23:03

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