I want to determined whether the unique irreducible $2$-dimensional representation $\rho$ of $Q_{8}$ realize over a given number field $K$.
My guess is that $\rho$ realize over $K$ if and only if $-1$ is a sum of two squares in $K$. My guess is from the statement of Exercise 12.3 in "Linear Representation of Finite Groups"(Serre). The point of the statement is that $K\left[Q_{8}\right]$ is quasi-split if and only if $-1$ is a sum of of two squares in $K$. Does this statement in the book implies my guess? If the statement does not, could you tell me whether my guess is right or not and recommend a reference related with my geuss?
Look at the $4$-dimensional $K$-algebra $\Bbb{H}_K= \{ a+bi+cj+dij,a,b,c,d\in K\},i^2=j^2=-1,ij=-ji$. It acts as a $\Bbb{H}_K$-module on itself.
Let $v=a+bi+cj+dij\ne 0\in \Bbb{H}_K,v^*=a-bi-cj-dij$. If $v^*v=a^2+b^2+c^2+d^2\ne 0$ then $\Bbb{H}_K.v=\Bbb{H}_K.1$. If for all $v, v^*v\ne 0$ then $\Bbb{H}_K$ is irreducible.
Otherwise take some $v\ne 0$ such that $v^*v=0$, then $\Bbb{H}_K.v$ has dimension $< 4$ thus (from our knowledge of the decomposition of $\Bbb{H_C}$) it is irreducible of dimension $2$.
$a^2+b^2+c^2+d^2=0$ means that $-1$ is a sum of 3 squares. If it is then let $w=b+ai+dj-cij$. We have $vw=0+Bi+Ci+Dij$ and $(vw)^*vw=w^*v^*vw=0\implies B^2+C^2+D^2=0$ ie. $-1$ is a sum of $2$ squares.