Calculating Radical of an Ideal [closed]
Solution 1:
For $I = (x^3, xy^5)$ (say in $k[x,y]$ for some field $k$), you want to calculate the radical $\sqrt{I}$ defined by $$\sqrt{I} = \{a \in k[x,y]: a^n \in I \text{ for some $n$}\}$$ We claim that $\sqrt{I} = (x)$.
- On one hand, $(x) \subset \sqrt{I}$ since $x^3 \in I$.
- On the other hand, if $a^n \in I$, clearly $a^n$ is a multiple of $x$. But as $x$ is a prime element in $k[x,y]$, we see that $a$ is already a multiple of $x$, i.e. $\sqrt{I} \subset (x)$.