Proposition

Prove assertion (ii) of Proposition 13. (Ideals, Varieties, and Algorithms) In other words, show that the least common multiple of two polynomials $f$ and $g$ in $k[x_1, \cdots, x_n]$ is the generator of the ideal $\ev{f} \cap \ev{g}$.

Solution

Suppose $\ev{h} = \ev{f} \cap \ev{g}$. Then $h \in \ev{f} \cap \ev{g}$, so $f \mid h$ and $g \mid h$.

On the other hand, suppose $f \mid p$ and $g \mid p$. Then $p \in \ev{f} \cap \ev{g} = \ev{h}$, so $h \mid p$.

By definition, $h = \lcm(f, g)$.