Let $I$ be an ideal in $\mathbb{C}[x_1 ,x_2 ,x_3 ,x_4 ]$ such that $I$ is generated by $x_1 x_3$, $x_2 x_3$, $x_1 x_4$, and $x_2 x_4$. How to show that this I cannot be generated by two elements?
It seems intuitively clear but, can't prove it rigorously
Note that $I$ contains no linear monomial, so the generating polynomials $p_i$ must be equal to $q_i+r_i$, where $q_i$ is a quadratic form and $r_i$ a sum of monomials of degree three or higher. Therefore, $x_1x_3$, $x_2x_3$, $x_1x_4$, and $x_2x_4$ are linear combinations of $q_i$'s, and so there are at least four $p_i$'s.
