3
$\begingroup$

The exercise is to find the field of fractions of the ring $k[x,y,z]/(xy^2-z^2)$ where $k$ is a field. I'm not exactly sure where to begin, and would appreciate some help/hints.

  • 0
    @rschwieb, since the ring of polynomials is a UFD, it's enough to see that $xy^2-z^2$ is irreducible---and this is clear since it is linear in $x$. Or you can check that it is a generator for the kernel of the map I defined in my answer below, implying that the quotient by the ideal it generates is a subring of a field (hence, a domain).2012-11-01

2 Answers 2

1

Start with the field of rational functions in 2 variables $F=k(t,u)$, and define a map $i:k[x,y,z] \rightarrow F$ by $i(x)=t^2/u^2$, $i(y)=u$, $i(z)=t$. This map identifies the quotient field of your ring with $F$.

0

Let $R:=k[x,y,z]/(xy^2-z^2)$, that is the elements of $R$ are (represented by) polynomials of variables $x,y,z$, but $R$ also satisfies the condition $xy^2-z^2=0 \ \text{ that is, }\ xy^2=z^2.$

The fraction field of $R$ consists of 'formal fractions' $\displaystyle\frac ab$ where $a,b\in R$ and of course $b\ne 0$ and $\displaystyle\frac{ca}{cb}=\frac ab$.

So, basically, two independent variables determine the third, say $y$ and $z$ are kept as independent, then, in the fraction field of $R$, we can substitute every $x$ by $x=z^2/y^2$.