# Characteristic of a field and algebraic curves

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!

Over characteristic $$p=3$$, $$X^3+Y^3+Z^3 = (X+Y+Z)^3$$.
• 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!! – mip Nov 26 '18 at 8:23