Math and stuff

Proposition Let $V$ be the twisted cubic in $\mathbb{R}^3$ and let $W = V(v - u - u^2)$ in $\mathbb{R}^2$. Show that $\phi(x, y, z) = (xy, z + x^2y^2)$ defines a polynomial mapping from $V$ to $W$. Solution $V = V(y - x^2, z - x^3) = \{ (a,...

Proposition Determine all solutions $(x, y) \in \mathbb{Q}^2$ of the system of equations \[\begin{align*} x^2 + 2y^2 &= 2, \\ x^2 + xy + y^2 &= 2. \end{align*}\] Also determine all solutions in $\mathbb{C}^2$. Solution Let $I = \ev{x^2 + 2y^2 - 2, x^2 + xy + y^2 - 2}$....

Proposition Show that the intersection of any collection of prime ideals is radical. Solution Let $\{ I_{\alpha} \}$ be a collection of prime ideals. Let $f \in k[x_1, \cdots, x_n]$. Suppose $f^m \in \bigcap_{\alpha} I_{\alpha}$ for some $m \geq 1$. For all $\alpha$, $f^m \in I_{\alpha}$ implies that $f \in...

Proposition Show that an ideal $I$ is prime if and only if for any ideals $J$ and $K$ such that $JK \subset I$, either $J \subset I$ or $K \subset I$. Solution Let ideals $J, K$ be given such that $J \not\subset I$ and $K \not\subset I$. Let $j \in...

Proposition Let $V, W \subset k^n$ be varieties. Prove that $I(V):I(W) = I(V \setminus W)$. Solution Let $f \in I(V):I(W)$. Then for all $g \in I(W), fg \in I(V)$. This implies that $fg$ vanishes on $V$ for all $g \in V$. Suppose that $f \notin I(V \setminus W)$. Then there...