The video I am getting this from is found here: https://www.youtube.com/watch?v=spHxUPvrkXw, it is around 5 minutes in.
The first part of the question is: for $F[X,Y] = Y^2 - X^3 = 0$ find $I(V(F[X,Y]))$.
Now the author says we have the inclusion $F[X,Y] \subset I(V(F[X,Y]))$ I don't understand how this works in that while I know that $V(F[X,Y])= \{X,Y \in K;F[X,Y]=0\}$ Where $K$ is the field, thus the vanishing set is just the set of zero's of the Polynomial $F[X,Y]$, but what is the "Ideal" of this? once I know that perhaps I will see why we have the inclusion above.
Perhaps someone could expound and tell me why the author pursues the following steps; the author goes on to say that this inclusion is really an equality, and he somehow gets this as he says we will take a polynomial $f \in I(V(F[X,Y]))g(X,Y) + a(X)Y + b(X)$ $p = (X= t^2,Y= t^3)$.
So I hope to understand
- What $I(V(F[X,Y])$ is and what it means
- Why $F[X,Y] = I(V(F[X,Y]))$
- What he is doing with the business of $f \in I(V(F[X,Y]))g(X,Y) + a(X)Y + b(X)$
Thank you,
Brian