Thread: irreducibility of a algebraic set

Show that the image of given by with (an algebraic set) is irreducible.

thanks

Originally Posted by roporte

Show that the image of given by with (an algebraic set) is irreducible.

thanks

an algebraic set is irreducible (or affine variety) iff is prime. so from this thread we only need to show that the ideal is a prime ideal of .

to see this define the map by: since is just a simple evaluation, it's a ring homomorphiam. also is onto since

for any finally observe that thus: now since is an integral domain, must be prime. Q.E.D.