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I'm trying to prove that $\overline{x}\in \frac{k[x,y,z]}{(xz-y^2)}$ is irreducible. (part of a problem 4.5 in Hulek's elementary algebraic geometry)

I attacked this by using the fact that every elt in $\frac{k[x,y,z]}{(xz-y^2)}$ has a unique representation of the form $f(x,z)y+g(x,z)$ and some elementary methods like comparing degree, resulting in a lot of pointless equations.

I want to complete the proof in this way, not using a kind of singularity or something hard(because TA said this method is valid), but I don't know how to.

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    Just to sanity check... did "some elementary methods" include assuming $x = uv$, writing $u = f(x,z) y + g(x,z)$ and $v = m(x,z) y + n(x,z)$, then trying to solve the equation $x = uv$ for $f,g,m,n$?2012-10-14
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    Yes, I tried in that way, but if there are any other way not using AG theory, it will be fine.2012-10-14
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    Maybe it helps to notice that $(xz-y^2)$ is a homogeneous ideal, and therefore the quotient ring is graded by degree.2012-10-14
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    Ah, thanks, Andrew. Maybe I spent a lot of time proving almost trivial proposition...2012-10-14

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