Let $\displaystyle I=<f_1,f_2> \subset \mathbb{C}[x,y]$ be an ideal generated by a linear and an irreducible quadratic polynomial.
Suppose that $\displaystyle g\in \sqrt{I}$. Show that $\displaystyle g^2\in I$
Is it not true that the definition of $\displaystyle \sqrt{I}$ is $\displaystyle \{ h | h^2 \in I \}$?
Thus, what you are asking is simply a definition.
Is it not true that the definition of $\displaystyle \sqrt{I}$ is $\displaystyle \{ h | h^2 \in I \}$?
no. the definition is: $\displaystyle \sqrt{I}=\{h | \ \exists n \in \mathbb{N}: \ h^n \in I \}.$ i don't like this problem very much, but i might think about it.