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Thread: Prove z^2-xy is irreducible.

  1. #1
    Nov 2010

    Prove z^2-xy is irreducible.

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

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

    is irreducible in $\displaystyle k[x,y,z]$.

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

    I have one proof, but it's basically brute force, using that if $\displaystyle f=gh$ then (WLOG) $\displaystyle \mathrm{deg}_x(g)=0$, 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
    Nov 2009
    Just a suggestion: you might be able to give some kind of geometric argument. If you decompose $\displaystyle f$, 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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