Find an ideal $\displaystyle I$ in $\displaystyle K[x,y]$ such that $\displaystyle V(I)$ consists of the point $\displaystyle (1,1)$ and the lines $\displaystyle x=0$ and $\displaystyle y=x+1$
Any help would be greatly appreciated...
Find an ideal $\displaystyle I$ in $\displaystyle K[x,y]$ such that $\displaystyle V(I)$ consists of the point $\displaystyle (1,1)$ and the lines $\displaystyle x=0$ and $\displaystyle y=x+1$
Any help would be greatly appreciated...
See this thread.
unfortunately tonio's answer is not correct because in his example we have $\displaystyle V(I)=\{(0,1)\}.$ recall that for any ideal $\displaystyle I$ of $\displaystyle K[x,y]$ we define $\displaystyle V(I)=\{p \in K^2: \ f(p)=0, \ \forall f \in I \}.$
anyway, to answer the question, you can choose $\displaystyle I=\langle x(y-x-1)(x-1), \ x(y-x-1)(y-1) \rangle.$ then it's easy to see that $\displaystyle V(I)$ is the set you're looking for.