Take two affine varieties in A^2(Q)
V_1 = {(x,y): x^2 = y^3}
V_2 = {(u,v) : u^3 = y^4}
with function fields Q(V_1) and Q(V_2)
a.) Show the function fields are isomorphic as Q-algebras
b.) Construct an explicit Birational map V_1 --> V_2 (dotted line)
Also,
Consider the lines
V_1={x=y=0}
V_2={y=z=0}
V_3={z=x=0}
Show that the product Ideal I(V_1)I(V_2)I(V_3) is smaller than the intersection of I(V_1), I(V_2), I(V_3), even though they define the same variety
b.) Suppose I,J \subset R are ideals such that I + J= R. Show that IJ = the intersection of I and J.