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Math Help - Prove z^2-xy is irreducible.

  1. #1
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    Prove z^2-xy is irreducible.

    I was wondering if anyone could think of an elegant proof that the polynomial

    f(x,y,z)=z^2-xy

    is irreducible in k[x,y,z].

    (My problem had \mathrm{char}k\neq 2, but I don't think it's relevant for this part.)

    I have one proof, but it's basically brute force, using that if f=gh then (WLOG) \mathrm{deg}_x(g)=0, and just writing out what the factors look like. I was wondering if there was a... nicer way.
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  2. #2
    Senior Member roninpro's Avatar
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    Just a suggestion: you might be able to give some kind of geometric argument. If you decompose f, then it would have to be into linear terms, which are hyperplanes. You could try to claim that your surface cannot be the product of hyperplanes.
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