Proof of Witt's result about quaternion extensions

Question

I'm searching for a proof of Witt's result that a biquadratic extension $K(\sqrt{a},\sqrt{b})/K$ extends to a Galois extension $L/K$ with quaternion group $Q_8$ iff the quadratic forms $<a,b,\frac{1}{ab}>, <1,1,1>$ are equivalent iff $(a,b)(a,a)(b,b) = 0 \in Br(K)$.

I know there is a proof in his original paper "Konstruktion von galoisschen Körpern der Charakteristik p zu vorgegebener Gruppe der Ordnung pf". However, I am searching for a proof in English. I would also like to find proofs from a more modern perspective as well as generalizations of the result.

Thank you.

Answer 1

A complete proof can be found in the first few pages of

https://mathscinet.ams.org/mathscinet-getitem?mr=977759

Jensen, Christian U.(DK-CPNH); Yui, Noriko(3-TRNT) Quaternion extensions. Algebraic geometry and commutative algebra, Vol. I, 155–182, Kinokuniya, Tokyo, 1988.

I will scan the relevant pages and post them here as soon as electricity is restored in my office.

Addendum: Here is the paper :

https://drive.google.com/open?id=0B8EHtI8F9qdIODNpZTQ1c1JjQTlIUUdsWm1nZmxRWVNTM1Bj

Answer 2

The problem you raise is tied to the "embedding problem." There is chapter devoted to this in Malle and Matzat's "Inverse Galois Theory" text (here's a link to the publisher's page about this book).

There's a summary of the embedding problem when the kernel is $\mathbb{Z}/2\mathbb{Z}$ in the paper by Klueners and Malle here (see subsection 3.4). Their Proposition 7 is a "Hasse principle" for embedding problems with kernel $\mathbb{Z}/2\mathbb{Z}$, and they give the example that

"If $L/\mathbb{Q}$ is a Galois extension with Galois group $(\mathbb{Z}/2\mathbb{Z})^{2}$, then $L$ is embeddable into a $Q_{8}$ extension if and only if $L$ is totally real and all odd primes $p$ that are ramified in $L$ have the property that $p \equiv 1 \pmod{4} \iff p$ has odd inertia degree in $L$."

It's not obvious to me that this is equivalent to Witt's statement in the case that $K = \mathbb{Q}$, but I can imagine it should be possible to sort out a clear connection in the context of cohomology (since the Brauer group of $K$ is $H^{2}(K,G_{m})$, and group extensions are also classified by elements of $H^{2}$).