Math and stuff

Proposition If $h \in k[x_1, \cdots, x_n]$ has total degree $N$ and if $\displaystyle F(t_1, \cdots, t_n) = \big(\frac{f_1(t_1, \cdots, t_m)}{g_1(t_1, \cdots, t_m)}, \cdots, \frac{f_n(t_1, \cdots, t_m)}{g_n(t_1, \cdots, t_m)}\big)$, show that $(g_1 \cdots g_n)^N(h \circ F)$ is a polynomial in $k[t_1, \cdots, t_m]$. Solution If $h$ is not, then $(g_1...

Proposition Let $k$ be an infinite field. Show that any straight line in $k^n$ is irreducible. Prove that any linear subspace of $k^n$ is irreducible. Solution 1 Let $a = (a_1, \cdots, a_n), b = (b_1, \cdots, b_n)$ be two distinct lines on the given line. Then for every $t...

Proposition Let $C$ be the curve in $k^2$ defined by $x^3 - xy + y^2 = 1$ and note that $(1, 1) \in C$. Now consider the straight line parametrized by \[\begin{align*} x &= 1 + ct, \\ y &= 1 + dt. \end{align*}\] Compute the multiplicity of this line...

Proposition If $P \subset k[x_1, \cdots, x_n]$ is a prime ideal, then prove that $P:f = P$ if $f \notin P$ and $P:f = \ev{1}$ if $f \in P$. Solution Suppose $f \notin P$. Let $g \in P:f$. Then $g\ev{f} \subset P$, so $gf \in P$. Since $P$ is prime,...

Proposition Theorem: if $k$ is an algebraically closed field, then every maximal ideal of $k[x_1, \cdots, x_n]$ is of the form $\ev{x_1 - a_1, \cdots, x_n - a_n}$ for some $a_1, \cdots, a_n \in k$. Prove that the theorem above implies that the Weak Nullstellensatz. Solution Let $I \ne k[x_1,...