I have the following two affine algebraic sets:
(i) $V_1=V(XYZ-Y^3,Z^3-X^5)$
(ii) $V_2=V(X^2+Y^2-1,X^2-Z^2-1).$
I want to find the irreducible components of them. I apologize for not posting any of my attempts, but I didn't make any significant progress. I just know that I have to write any of the sets $V_1,V_2$ as the union of a finite number of affine algebraic sets and then, I know that I should prove that these algebraic sets are irreducible by proving that the ideal that generates them is prime.
Can you please show me how we work in at least one of the two cases above?
Thanks in advance for your help!