# Math Help - irreducibility of a algebraic set

1. ## irreducibility of a algebraic set

Show that the image of $\phi : K \rightarrow K^3$ given by $\phi(t) = (t,f(t),g(t))$ with $f,g \in K[x]$ (an algebraic set) is irreducible.

thanks

2. Originally Posted by roporte
Show that the image of $\phi : K \rightarrow K^3$ given by $\phi(t) = (t,f(t),g(t))$ with $f,g \in K[x]$ (an algebraic set) is irreducible.

thanks
an algebraic set $X$ is irreducible (or affine variety) iff $\mathcal{I}(X)$ is prime. so from this thread we only need to show that the ideal $J=$ is a prime ideal of $K[x,y,z]$.

to see this define the map $\varphi: K[x,y,z] \longrightarrow K[x]$ by: $\varphi(h(x,y,z))=h(x,f(x),g(x)).$ since $\varphi$ is just a simple evaluation, it's a ring homomorphiam. also $\varphi$ is onto since $\varphi(h(x))=h(x),$

for any $h(x) \in K[x].$ finally observe that $\ker \varphi = J.$ thus: $\frac{K[x,y,z]}{J} \simeq K[x].$ now since $K[x]$ is an integral domain, $J$ must be prime. Q.E.D.