Math and stuff

Proposition Show that in $\mathbb{Q}[x, y]$, we have \[\begin{align*} \ev{(x + y)^4(x^2 + y)^2(x - 5y)} \cap \ev{(x + y)(x^2 + y)^3(x + 3y)} = \ev{(x + y)^4(x^2 + y)^3(x - 5y)(x + 3y)}. \end{align*}\] Solution By Proposition 13 on P.195 (Ideals, Varieties, and Algorithms), it suffices to find the...

Proposition Show that $\ev{x^2 + 1} \subset \mathbb{R}[x]$ is a radical ideal, but that $V(x^2 + 1)$ is the empty variety. Solution $x^2 + 1 \geq 1$ for each $x \in \mathbb{R}$, so $V(x^2 + 1)$ is empty. Since $x^2 + 1$ has no roots in $\mathbb{R}$, $x^2 + 1$...

Proposition Let $f$ and $g$ be distinct non-constant polynomials in $k[x, y]$ and let $I = \ev{f^2, g^3}$. Is it necessarily true that $\sqrt{I} = \ev{f, g}$? Solution No. Let $f = x^3, g = x^4$. Then $I = \ev{x^6, x^{12}}$. Clearly, $x \in \sqrt{I}$. For every $h_1, h_2 \in...

Proposition Given a field $k$, show that $\sqrt{\ev{x^2, y^2}} = \ev{x, y}$, and, more generally, show that $\sqrt{\ev{x^m, y^n}} = \ev{x, y}$ for any positive integers $n$ and $m$. Solution We have $x, y \in \sqrt{\ev{x^m, y^n}}$. Since the radical of an ideal is an ideal by Lemma 5 (P....

Proposition $\mathbf{V}(y - x^2, z - x^3)$ is the twisted cubic in $\mathbb{R}^3$. Show that $\mathbf{V}((y - x^2)^2 + (z - x^3)^2)$ is also the twisted cubic. Show that any variety $\mathbf{V}(I) \subset \mathbb{R}^n, I \subset \mathbb{R}[x_1, \cdots, x_n]$, can be defined by a single equation. Solution 1 A point...