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In what way the characteristic of a field influences the curves defined over that field? For example, let $C$ be the curve defined by $X^3+Y^3+Z^3=0$ over an algebraically closed field $K$. What happen to the curve if $char(K)=3$ or $char(K)\ne 3$? Thank you!

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Over characteristic $p=3$, $X^3+Y^3+Z^3 = (X+Y+Z)^3$.

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  • $\begingroup$ Can you explain why is that true? $\endgroup$ – mip Nov 26 '18 at 8:12
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    $\begingroup$ @mip Why not just multiply out the cube and see what happens? $\endgroup$ – Mark Bennet Nov 26 '18 at 8:16
  • $\begingroup$ Oh, I see! Besides $X^3, Y^3$ and $Z^3$, all other terms from the expansion have a coefficient divisible by 3, which mean that they are $0$. Thank you!! $\endgroup$ – mip Nov 26 '18 at 8:23

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