For distinct primes p and q that are $3$ mod $4$, the quaternion algebras over the rationals, $(−1, p)_\mathbb{Q}$ is not isomorphic to $(−1, q)_\mathbb{Q}$.
Any help prove it? I'm able to show that $(−1, q)_\mathbb{Q}$ is a division ring, for a prime $q =$ 3 mod 4
There is an important norm criterion for the isomorphism of quaternion algebras with a common slot (outside characteristic $2$): for a field $F$ not of characteristic $2$, and nonzero $a, b, b'$ in $F$, we have $(a,b)_F \cong (a,b')_F$ if and only if $b/b' = x^2 - ay^2$ for some $x$ and $y$ in $F$. (This is roughly analogous to $F(\sqrt{d}) \cong F(\sqrt{d'})$ if and only if $d/d' = x^2$ for some $x \in F$, where $d$ and $d'$ are nonsquares in $F$.)
If you don't know this criterion, read up on it. If you know it, then apply it.