Math and stuff

Proposition If $I \subset k[x_1, \cdots, x_n]$ is an ideal, prove that $\ev{ \LT(g) \mid g \in I \setminus \{ 0 \} } = \ev{ \LM(g) \mid g \in I \setminus \{ 0 \} }$. Solution Let $T = \ev{ \LT(g) \mid g \in I \setminus \{ 0 \} },...

Proposition Let $I$ be an ideal in $k[x_1, \cdots, x_n]$. Prove that $1 \in I$ if and only if $I = k[x_1, \cdots, x_n]$. More generally, prove that $I$ contains a nonzero constant if and only if $I = k[x_1, \cdots, x_n]$. Suppose $f, g \in k[x_1, \cdots, x_n]$ satisfy...

Proposition Let $I$ be an ideal in $k[x_1, \cdots, x_n]$, where $k$ is an arbitrary field. Show that $\sqrt{I}$ is a radical ideal. Show that $I$ is radical if and only if $I = \sqrt{I}$. Show that $\sqrt{\sqrt{I}} = \sqrt{I}$. Solution 1 First, we will show that $\sqrt{I}$ is an...

Proposition Prove that an algebraically closed field $k$ must be infinite. Solution Suppose $k = \{ a_1, \cdots, a_n \}$ is finite. Then $f(x) = 1 + \prod_{i=1}^{n} (x - a_i)$ is a polynomial such that $f(a_i) = 1 \ne 0$ for each $i$. Thus $f$ has no roots in...

Proposition The intersection $I \cap J$ of two principal ideals $I = \ev{f}, J = \ev{g} \subset k[x_1, \cdots, x_n]$ is a principal ideal. Solution Let $h = \lcm(f, g)$. Then $f \mid h$ and $g \mid h$, so $\ev{h} \subset \ev{f}$ and $\ev{h} \subset \ev{g}$. Thus $\ev{h} \subset \ev{f}...