Math and stuff

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}...

Proposition Show that $I(\{a_1, \cdots, a_n\}) = \ev{x_1 - a_1, \cdots, x_n - a_n}$. Solution Let $I = I(\{a_1, \cdots, a_n\})$. Since each $x_i - a_i$ vanishes at $\{a_1, \cdots, a_n\}$, $x_i - a_i \in I$ for each $i$. Thus $\ev{x_1 - a_1, \cdots, x_n - a_n} \subset I$. $\ev{x_1...

Proposition Let $I, J$ be ideals in $k[x_1, \cdots, x_n]$ and suppose that $I \subset \sqrt{J}$. Show that $I^m \subset J$ for some integer $m > 0$. Solution By the Hilbert Basis Theorem, $I = \ev{f_1, \cdots, f_l}$ for some $f_i$. Since $I \subset \sqrt{J}$, we have $m_i \geq 1$...

Proposition Show that $N$ can be arbitrarily large in $I:J^{\infty} = I:J^N$. Solution Let $N \in \mathbb{N}$ be given. Let $I = \ev{x^N(y - 1}, J = \ev{x}$. Then clearly, $I:J^{\infty} = I:J^N$. Then $y - 1 \in I:J^{\infty}$ because $\forall a \in J, (y - 1)a^N \in I$. Let...

Proposition For an arbitrary field, show that $\sqrt{IJ} = \sqrt{I \cap J}$. Give an example to show that the product of radical ideals need not be radical. Also give an example to show that $\sqrt{IJ}$ can differ from $\sqrt{I}\sqrt{J}$. Solution Since $IJ \subset I \cap J$, $\sqrt{IJ} \subset \sqrt{I \cap...