Show that $\phi : K \rightarrow K^3$ given by $\phi(t) = (t,f(t),g(t))$ with $f,g \in K[x]$ is an algebraic set of $K^3$
Show that $\phi : K \rightarrow K^3$ given by $\phi(t) = (t,f(t),g(t))$ with $f,g \in K[x]$ is an algebraic set of $K^3$
let $p(x,y,z)=y -f(x), \ q(x,y,z)=z-g(x).$ clearly both $p(x,y,z)$ and $q(x,y,z)$ are in $K[x,y,z]$ and the image of $\phi$ is the solutions of the system: $p(x,y,z)=0, \ q(x,y,z)=0. \ \ \Box$