Radical ideal

**KaKa** · January 14th 2009, 03:04 AM

Let $I=<f_1,f_2> \subset \mathbb{C}[x,y]$ be an ideal generated by a linear and an irreducible quadratic polynomial.

Suppose that $g\in \sqrt{I}$ . Show that $g^2\in I$

**ThePerfectHacker** · January 14th 2009, 10:06 AM

Originally Posted by KaKa

Let $I=<f_1,f_2> \subset \mathbb{C}[x,y]$ be an ideal generated by a linear and an irreducible quadratic polynomial.

Suppose that $g\in \sqrt{I}$ . Show that $g^2\in I$

Is it not true that the definition of $\sqrt{I}$ is $\{ h | h^2 \in I \}$ ?
Thus, what you are asking is simply a definition.

**NonCommAlg** · January 14th 2009, 10:07 PM

Originally Posted by ThePerfectHacker

Is it not true that the definition of $\sqrt{I}$ is $\{ h | h^2 \in I \}$ ?

no. the definition is: $\sqrt{I}=\{h | \ \exists n \in \mathbb{N}: \ h^n \in I \}.$ i don't like this problem very much, but i might think about it.

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