Show that the image of $\displaystyle \phi : K \rightarrow K^3$ given by $\displaystyle \phi(t) = (t,f(t),g(t))$ with $\displaystyle f,g \in K[x]$ (an algebraic set) is irreducible.
thanks
an algebraic set $\displaystyle X$ is irreducible (or affine variety) iff $\displaystyle \mathcal{I}(X)$ is prime. so from this thread we only need to show that the ideal $\displaystyle J=<y-f(x), z - g(x)>$ is a prime ideal of $\displaystyle K[x,y,z]$.
to see this define the map $\displaystyle \varphi: K[x,y,z] \longrightarrow K[x]$ by: $\displaystyle \varphi(h(x,y,z))=h(x,f(x),g(x)).$ since $\displaystyle \varphi$ is just a simple evaluation, it's a ring homomorphiam. also $\displaystyle \varphi$ is onto since $\displaystyle \varphi(h(x))=h(x),$
for any $\displaystyle h(x) \in K[x].$ finally observe that $\displaystyle \ker \varphi = J.$ thus: $\displaystyle \frac{K[x,y,z]}{J} \simeq K[x].$ now since $\displaystyle K[x]$ is an integral domain, $\displaystyle J$ must be prime. Q.E.D.