2

I have a quotient polynomial ring $ R = k[X,Y,Z]/ \langle X^2 - Y^3-1, XZ-1 \rangle$ where $k$ is a field and $X,Y,Z$ are variables.

Let $x, y, z $ be the images of $X,Y,Z$ respectively. Fixing $a, b \in k$ and writing $ t = x +ay +bz$, I need to show that $x, y $ are integral over $P = k[t]$.

So I think $x = X + A(X^2-Y^3-1) + B(XZ-1) $ where $A, B \in k[X,Y,Z]$ with similar expressions for $y, z$. But I am not sure about anything else. I am sorry I do not have more to show for my work. Thanks.

Martin Sleziak
  • 53,687
Vishesh
  • 2,928

1 Answers1

1

We have $x^2=y^3+1$ and $zx=1$. We might as well write $1/x$ for $z$. Then $t=x+ay+b/x$ so $$ay=t-x-\frac bx.$$ Then $$a^3x^2=a^3y^3+a^3=\left(t-ax-\frac bx\right)^3+a^3.$$ Multiplying by $x^3$ gives $$a^3x^5=(tx-ax^2-b)^3+a^3x^3.$$ If $a\ne0$ this equation can be rewritten as $$a^3x^6+\textrm{ lower terms in }x, t$$ which, when we divide by $a^3$ gives $x$ as integral over $k[t]$.

If $a=0$ we get $t=x+a/x$ and then $$x^2-tx=a=0$$ so still $x$ integral over $k[t]$.

As $y$ is integral over $k[x]$ then $y$ is integral over $k[t]$.

Angina Seng
  • 158,341