1. ## Algebraic Geometry

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.

2. Originally Posted by terr13

b.) Suppose I,J \subset R are ideals such that I + J= R. Show that IJ = the intersection of I and J.
well, i will assume that R is a commutative ring..

note that $I\cap J \subset J$ and $I\cap J \subset I$

also, $I\cdot R = I$ for any ideal $I$ of R.

thus $I\cap J = (I\cap J) \cdot R = (I\cap J) \cdot (I+J) = (I\cap J) \cdot I + (I\cap J) \cdot J \subset JI + IJ = IJ$

for the other direction, if $a \in IJ$, then

$a = \sum x_iy_i$ (finite sum) where $x_i \in I$ and $y_i \in J$

properties of ideals will show you that $a$ is also in $I\cap J$