All Questions

Prime ideals in $\mathbb{C}[x,y]$ were listed here; they are: (i) $(0)$. (ii) $(f)$, where $f$ is an irreducible polynomial. (iii) $(x-\lambda,y-\mu)$, where $\lambda,\mu \in \mathbb{C}$. Now let $R \...

Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_0, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...

Fix $\ell \geq 3$, $r \geq 2$ and $1 \leq k \leq \ell - 1$ and $z_1, \ldots, z_\ell \in \mathbb{C}$ with $z_i \neq 0$ for all $i$ and $z_i \neq z_j$ for all $i \neq j$. Now consider the (irreducible, ...

Let $f,g$ be two polynomials in $\mathbb{Z}[x,y]$, given by $$ f(x,y)=x^4-3xy+y^2,$$ $$ g(x,y)=x^5-4xy+3xy^2.$$ Let $I=(f,g)$ be the ideal in $\mathbb{Z}[x,y]$ generated by $f$ and $g$. Is $x,x^2,...

Im trying to find a characterization of the prime ideals in the polynomial ring $R = \mathbb Z[X,Y]$ in two variables over the integers. Actually I need to find the maximal ideals in quotient rings $...