I'm studying the book "Twenty-Four Hours of Local Cohomology" and here is an exercise I'm stuck in.
Consider the map $\phi:K[u,v,w,x,y,z]\to K[ar,br,cr,as,bs,cs]$ where $K$ is a field and $$\phi(u)=ar,\phi(v)=br,\phi(w)=cr$$ $$\phi(x)=as,\phi(y)=bs,\phi(z)=cs.$$ Prove that $\ker \phi=(vz-wy,wx-uz,uy-vx)$ and conclude that this is a prime ideal.
Before that there is an exercise that I think the authors want me to use on this problem
"Let $R$ and $S$ be algebras over a field $K$ and $\phi:R\to S$ a surjective $K$-algebra homomorphism. Let $\{s_i\}$ be a $K$-vector space basis for $S$ and $r_j\in R$ s.t. $\phi(r_j)=s_j$ for all $j$. Let $\mathfrak{a}$ be an ideal contained in $\ker \phi$. If every element of $R$ is congruent to an element in the $K$-span of $\{r_j\}$ modulo $\mathfrak{a}$, prove that $\mathfrak{a}=\ker \phi$."
This exercise is very direct, in my opinion, but only because we have strong hypothesis, especially the last one. I think if I want to apply this, I have to list all $s_i$ and $r_i$, which is quite hard since we are talking about $K$-span, so there are too many possibilities here (degree 2 only needs about 15 elements).
On the other hand, if I want to do it directly: take a polynomial and take modulo $\mathfrak{a}$, then I'm stuck here since there are too many variables (6) and hence many possibilities. Sure, if I'm more careful I should be able to get it done but too many monomials really confuse me.
Of course other approaches are also appreciated.