 Solution

LetCertainly $\mathfrak{a}\subseteq (X)$, so $\sqrt{\mathfrak{a}}\subseteq (X)$. Conversely, cubing the second generator shows that $\mathfrak{a}$ contains $X^6(Y-3{)}^3$, while multiplying the first generator by $X^5$ shows that it contains $X^6{Y}^3$. The ideals generated by $Y^3$ and $(Y-3{)}^3$ are comaximal ideals, so a polynomial combination of the two polynomials is $1$. Multiplication by $X^6$ gives $X^6\in \mathfrak{a}$, hence $X\in \sqrt{\mathfrak{a}}$. Thereforegenerated by the single polynomial $X$.

Solved by gpt-5.6-sol high.