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.
Finding the field of fractions of $k[x,y,z]/(xy^2-z^2)$
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commutative-algebra
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0Alternatively, you can use that localization commutes with quotients. – 2012-10-18
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0@all So, out of curiosity, what's the trick to instantly spot that $(xy^2-z^2)$is a prime ideal? – 2012-10-18
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0Actually the exercise stated that you may assume that the ring is an integral domain, I guess I should have mentioned that. But I'd love to see some trick for actually proving this! – 2012-10-18
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0@rschwieb, this probably needs more justification, but it looks like we need a square root for $x$ in order to factor $xy^2-z^2.$ – 2012-10-19
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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