# Unique multiplicative quadratic form on quaternion algebras

I want to prove, that the only multiplicative quadratic form $$Q$$ (so $$Q(xy)=Q(x)Q(y) \forall x,y$$) on a quaternion algebra $$\Big(\dfrac{a,b}{F}\Big)$$ is the norm $$\mathrm{Nr}$$, which is isometric to the form $$\langle 1,-a,-b,ab\rangle$$ in the orthogonal basis $$1,i,j,ij$$. One easily sees, that $$Q(1)=Q(1^2)=Q(1)^2\Rightarrow Q(1)=1$$ and similarly $$Q(i)^2=a^2$$, $$Q(j)^2=b^2$$ and therfore $$Q(ij)=(ab)^2$$. But how do we show $$Q(i)=-a$$ and $$Q(j)=-b$$? Or is it even necessary? It seems like I'm missing something obvious.

• A couple of things to keep in mind: (1) your argument that $Q(1) = 1$ is only valid if you assume $Q(1) \ne 0$, and (2) in general, a quadratic form is not determined by its values on a basis. – Kimball Jan 24 at 22:06