I have been struggling with this question 1.25 of Fulton because I cannot find an example of someone doing this explicitly.
I am asked to decompose: $V (Y^4 - X^2,Y^4- X^2Y^2 +XY^2-X^3)$ into irreducible factors.
We see that: $$ (Y^2-X)(Y^2+X)=Y^2-XY^2+XY^2-X^2=Y^4-X^2.$$ $$(Y+X)(Y-X)(Y^2+X)=Y^4- X^2Y^2 +XY^2-X^3.$$
We then use the properties from Fulton to simplify the algebraic sets.
$$V (Y^4 - X^2,Y^4- X^2Y^2 +XY^2-X^3)$$ $$=V((Y+X)(Y-X)(Y^2+X), (Y^2-X)(Y^2+X)).$$ We split off the common factor. $$V( (Y^2+X) ((Y+X)(Y-X), (Y^2-X))= $$ $$V(Y^2+X) \cup V((Y+X)(Y-X), (Y^2-X)) $$ We simplify the last bit. $$V((Y+X)(Y-X), (Y^2-X))= V((Y+X)(Y-X)) \cap V(Y^2-X) $$ $$= (V(Y+X) \cup V(Y-X)) \cap V(Y^2-X). $$ We use the distributive law of unions and intersections and arrive at: $$V((Y+X)(Y-X), (Y^2-X))= $$ $$ (V(Y+X) \cap V(Y^2-X)) \cup (V(Y-X) \cap V(Y^2-X)$$ Which in the end gets us: $$V(Y^2+X) \cup (V(Y+X) \cap V(Y^2-X)) \cup (V(Y-X) \cap V(Y^2-X)$$ As a decomposition into irreducible factor.... but this does not look very satisfactory, is this okay? I have written it as a union of objects, but how do I know these are indeed irreducible?