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I'm trying to solve the following exercise from Professor Vakil's book:

3.6.F. TRICKY EXERCISE. (a) Suppose $I = (wz−xy, wy−x^2 , xz−y^2) \subset k[w, x, y, z]$. Show that $\operatorname{Spec} k[w, x, y, z]/I$ is irreducible, by showing that $k[w, x, y, z]/I$ is an integral domain. (This is hard, so here is one of several possible hints: Show that $k[w, x, y, z]/I$ is isomorphic to the subring of $k[a, b]$ generated by monomials of degree divisible by 3.)

I don't really know where to start, even with the hint. I can see that $w$, $x$, $y$, and $z$, will all be nonzero in the quotient. Also there will be $wx$, $w^2$, $zy$, and $z^2$. These are all of degree 1 or 2, and they cannot be written as $a^3$, so I don't think I understand the Professor's hint.

Thanks!!!

user26857
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solvergirl
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  • How do you show the quotient ring is a domain? 2. What does it mean for Spec to be irreducible? What is the nilradical of the quotient? Also, its spec not speck.
    • – Prince M Feb 14 '17 at 09:49
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    @PrinceM: 1) That's the question :o 2) I don't understand. The irreducibility follows from k[w,x,y,z]/I being a domain. That's the part I need help with. – solvergirl Feb 14 '17 at 09:58
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    @PrinceM It's not Speck, it's Spec k where the space disappeared because it always does in math mode. – Arthur Feb 14 '17 at 10:04
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    Don't focus too much on $k[x,y,z,w]/I$. Instead, try to construct a homomorphism $k[x,y,z,w]\to k[a,b]$ where the image is the given subring, and the kernel is $I$. – Arthur Feb 14 '17 at 10:08
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    http://math.stackexchange.com/a/227255/121097 – user26857 Feb 14 '17 at 12:16
  • Related: https://math.stackexchange.com/questions/142490 – Watson Feb 17 '17 at 15:20