Let $q$ be the anisotropic,quadratic form of rank two corresponding to $\alpha = d(q) \in H^1(k,\mu_2)$. In his lecture notes "Topics in quadratic Forms" Vishik writes:
For $n=1$ we get the Rost-Motive $M_\alpha = M(\mathrm{Spec}(k(\sqrt{d})))$.
I want understand how to see this.
- Obviously $M_\alpha$ splits into two trival Tate-Motives over $k(\sqrt{d})$.
- From the injection of $k$ into $k(\sqrt{d})$, one gets a surjective cover on the spectra, which is branched at $d$ with order two. So one has vage premonition that the indecomposable $M_\alpha$ involves $\mathrm{Spec}(k(\sqrt{d}))$ somehow. But how exactly? And how in respect to the theory of algebraic cycles?