Math and stuff

Proposition Let $I$ and $J$ be ideals in $k[x_1, \cdots, x_n]$. Show that if $I$ is radical, then $I:J$ is radical and $I:J = I:\sqrt{J} = I:J^{\infty}$. Solution Let $f \in k[x_1, \cdots, x_n]$ and suppose $f^m \in I:J$ for some $m \geq 1$. Then $\forall g \in J, f^mg...

Proposition Show that a prime ideal is radical. Solution Let $P$ be a prime ideal. Let $m = 1$. Let $f$ be given such that $f^m \in P$. Then $f \in P$. Suppose that $f^m \in P \implies f \in P$ for all $f$ for some $m \geq 2$. Let...

Proposition Prove that $I$ and $V$ are inclusion-reversing and $V(\sqrt{I}) = V(I)$ for any ideal $I$. Solution Let $V_1 \subset V_2$. Then $I(V_2) \subset I(V_1)$ because every polynomial that vanishes on $V_2$ must vanish on $V_1$. Similarly, if $I_1 \subset I_2$, then $V(I_2) \subset V(I_1)$ because if every polynomial in...

Proposition Consider the system of equations \[\begin{align*} x^2 + 2y^2 &= 3, \\ x^2 + xy + y^2 &= 3. \end{align*}\] If $I$ is the ideal generated by these equations, find bases of $I \cap k[x]$ and $I \cap k[y]$. find all solutions of the equations. Which of the solutions...

Proposition Let $I \subset k[x_1, \cdots, x_n]$ be an ideal. Prove that $I_l = I \cap k[x_{l + 1}, \cdots, x_n]$ is an ideal of $k[x_{l + 1}, \cdots, x_n]$. Prove that the ideal $I_{l + 1} \subset k[x_{l + 2}, \cdots, x_n]$ is the first elimination ideal of $I_l...