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Let $X=Z(f_1, f_2, ......, f_n)$ be the algebraic set defined by some polynomials $f_i\in\mathbb{C}[x_1, x_2, ....., x_n]$. Show that $X$ only depends on the ideal generated by $f_i$; that is $X=Z(I)$ where $I=(f_1, f_2, ......, f_r)$.

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Certainly $Z(I)\subseteq X$, as $f_1,\ldots,f_n\in I$ so if $f(x)=0$ for all $f\in I$ then $f_1(x)=\cdots=f_n(x)=0$. If $x\in X$ and $f\in I$, then we can write $f=g_1f_1+\cdots+g_nf_n$ for some $g_1,\ldots,g_n\in \mathbb C[x_1,\ldots,x_n]$ and so $f(x)=g_1(x)f_1(x)+\cdots+g_n(x)f_n(x)=0$ hence $x\in Z(I)$, thus $X\subseteq Z(I)$. This establishes $X=Z(I)$.

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