i'll assume that the base field is

define by and

let and also for any let define the map2. Show that the circle x^{2}+y^{2}=1 is isomorphic (as an affine variety)

to the hyperbola xy=1.

by: now if then for some

but then: i.e. so is well-defined.clearly is a surjective

ring homomorphism. now suppose then i.e. for some therefore:

so hence so is injective and we're done!

1) and are not isomorphic because is not a UFD:but neither is isomorphic to A^{1}.

2) and are not isomoprphic because, if there was an isomorphism then assuming that

we'll get thus and therefore

are constant. but then would not be surjective. Q.E.D.