I am looking for the reference to Hasse's original publication classifying the Quaternion algebras over a number field $K$. The statement in the paper or book should read something like:

``Two Quaternion algebras over a number field $K$ are isomorphic iff they are ramified at the same number of places"

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    $\begingroup$ Ramification is for quaternion algebras over number fields, not over arbitrary fields? $\endgroup$ – Dietrich Burde Feb 1 at 15:11
  • $\begingroup$ oops corrected the question $\endgroup$ – Sam Hughes Feb 1 at 15:12
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    $\begingroup$ Theorem 4.8. here says:"Let $H$ and $H'$ be quaternion algebras over a number field $F$. Then $H\cong H'$ if and only if $Ram(H) = Ram(H')$. $\endgroup$ – Dietrich Burde Feb 1 at 15:15
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    $\begingroup$ It says $H_v = H \otimes_K K_v$, or $H_v$ is a division algebra (and $v$ is said ramified) or $H_v \cong M_2(K_v)$. Theorem 1.13 says it is about the equation $ax^2+by^2=1, (x,y) \in K_v$ where $H=K[i,j], i^2=a,j^2=b,ij = -ji$ $\endgroup$ – reuns Feb 1 at 22:17

As detailed here, Hasse introduced an invariant, now known as the Hasse invariant in two papers (here and here) to classify central simple algebras.

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    $\begingroup$ In particular, for rank $4$ (quaternion algebras), the local invariant is either $0$ or $1/2$, the latter being the nonsplit (ramified) case. These local invariants must add up to $0\pmod{\Bbb Z}$. So to specify the isomorphism class it’s enough to specify which local invariants are $1/2$. $\endgroup$ – Lubin Feb 2 at 17:58

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