reference request - Galois groups via cohomology

I don't have a reference, but it does not seem too hard.

1. Assume $L/k$ normal, and take $sigma in G$, which can be extended to an element of $Gal(L/k)$. Then $sigma(sqrt{mu})/sqrt{mu} in L$, and if $gamma$ is the nontrivial element of $Gal(L/K)$, $sigma^{-1} gamma sigma$ is trivial on $K$, and being nontrivial on $L$ it has to be equal to $gamma$. So $gammaleft( sigma(sqrt{mu})/mu right) = sigma(gamma(sqrt{mu}))/gamma(sqrt{mu}) = sigma(sqrt{mu})/sqrt{mu}$, so we can take $alpha_{sigma} = sigma(sqrt{mu})/sqrt{mu}$, it is an element of $K$.
   Now for the other way, take $tilde{sigma} in Gal(bar{L}/k)$. Denote the restriction of $tilde{sigma}$ to $K$ by $sigma$. Then $sigma(sqrt{mu})/sqrt{mu}= pm alpha_{sigma}$ (this equality is in $bar{L}$), so $sigma(sqrt{mu}) = pm alpha_{sigma} sqrt{mu} in L$, so $L$ is normal.




2. Consider a set-theoretic section $sigma mapsto tilde{sigma}$ for the surjective morphism $Gal(L/k) rightarrow G$. Then the (up to a coboundary) 2-cocycle $beta_0$ associated to the group extension is given by the formula $tilde{sigma} tilde{tau} = beta_0(sigma,tau) widetilde{sigma tau}$. Evaluating at $sqrt{mu}$ gives the equality between $beta$ and $beta_0$, if for every $sigma in G$, $alpha_{sigma} = tilde{sigma}(sqrt{mu})/sqrt{mu}$. You can always choose your section so that it is the case (change of section = associating a sign to each $sigma in G$).

EDIT: There's a left/right action problem, because $beta_0(sigma,tau) = sigma(alpha_{tau}) alpha_{sigma} alpha_{sigma tau}^{-1}$ with what I wrote. I think it has to do with the fact that you use exponential notation, so somehow your action is on the right? Maybe you define $x^{sigma} = sigma^{-1}(x)$? Otherwise the definition of $beta$ doesn't make it a 2-cocycle, with the definitions I know. Could you clarify?