Proposition

Show that in $\mathbb{Q}[x, y]$, we have

\[\begin{align*} \ev{(x + y)^4(x^2 + y)^2(x - 5y)} \cap \ev{(x + y)(x^2 + y)^3(x + 3y)} = \ev{(x + y)^4(x^2 + y)^3(x - 5y)(x + 3y)}. \end{align*}\]

Solution

By Proposition 13 on P.195 (Ideals, Varieties, and Algorithms), it suffices to find the lowest common multiplier of the two generators, and the right hand side is the ideal generated by the two generators.