Field structure on $\mathbb{R}^2$ I have the following question:
Is there a simple way to see that if we put a multiplication $*$ on $\mathbb{R}^2$ (considered as a vector space over $\mathbb{R}$) such that with usual addition and this multiplication $\mathbb{R}^2$ becomes a field, then there exists a nonzero $(x,y)$ such that $(x,y)*(x,y)=-1$?
Remark:


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*What I mean by "a simple way to see" is that I really don't want to refer to Frobenius's Theorem on real finite dimensional division algebras.

*I haven't said this in the problem but I'm also assuming that with this multiplication $\mathbb{R}^2$ becomes an algebra meaning $x*(\alpha y)=\alpha(x*y).$

You are looking at a field extension of $\Bbb R$ of degree two, but by the fundamental theorem of algebra, that field sitting above $\Bbb R$ is unique up to isomorphism and is isomorphic to $\Bbb C$.
