For a field $K$ (we may assume it is infinite), I would like to give a parametric definition of the algebraic variety $V(x^2-y^2-1,x^2-z^2-1)\subset K^3$.
I was thinking that, if $K$ were for example $\mathbb{R}$, I could give this parametrization: $\{(\sqrt{t+1},\sqrt{t}),\sqrt{t}): t \in \mathbb{R}\}$. But I do not think I can take the square root in any field $K$, because I cannot assume it is algebraically closed. Also, even in $\mathbb{R}$, tha fact that I could have different signs for the square roots makes me think my parametrization is not correct.
Any help or hints would be really useful.