radical membership and ideal membership

Question

Consider the ideal $I=(x^3y-x^2y^2,x^3z+z^2yx,x^2-xz)\subset \Bbb Q[x,y,z].$ Is $x\in I?$ Is $x\in \sqrt I?$

I'm assuming a question like this is quite simple and that there is just a method, if possible could someone explain the method or direct me to a similar example of a worked solution.

Answer 1

Reposting my comments as an answer, to add a picture of worked example as you want:

Cox , Little, O'Shea 's book Ideals, Varieties, and Algorithms page 82, Corollary 2 has:

Let $G = \{g_t, \cdots, g_t\}$ be a Groebner basis for an ideal $I \subset k[x_1, . . . , x_n]$ and let $f \in k[x_1, . . . , x_n].$ Then $f \in I$ if and only if the remainder on division of $f$ by $G$ is zero.

To find Groebner basis for an ideal $I$ you can use CoCoA. Here is the result for $I$:

You can see an sketch of what to do, as "Example 1 at page 96" of the book.

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For 2nd, you can use Proposition 8 (radical membership algorithm), page 178 of book:

Let $k$ be an arbitrary field and let $I = (f_1, . . . , f_s) ⊂ k[x_1, . . . , x_n]$ be an ideal. Then $f \in \sqrt I$ if and only if the constant polynomial $1$ belongs to the ideal $\tilde I = (f_1, . . . , f_s , 1 − y f) ⊂ k[x_1, . . . , x_n, y].$

So we compute a reduced Groebner basis of the ideal $(f_1, . . . , f_s , 1 − y f)$. Then $f \in \sqrt I$ if and only if the result is $\{1\}.$
In the Picture, below, you can see that $x\in \sqrt I$: