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Corollary 2

[proposition] Let $G = \{g_1, \ldots, g_t\}$ be a Grobner basis for an ideal $I \subseteq k[x_1, \ldots, x_n]$ and let $f \in k[x_1, \ldots, x_n]$. Then $f \in I$ if and only if the remainder on division of $f$ by $G$ is zero. [/proposition]