Let $k$ be a field. On page 5 of Milne's Elliptic Curves, the author defines an algebraic curves to be defined by polynomials $f \in k[x,y]$ with no repeated irreducible factors in $\overline{k}[x,y]$.
Why does Milne require $f$ to have no repeated factors? The only reason I can think of is the following: a point on the intersection of two irreducible components is singular, so having repeated irreducible factors would introduce singularies.
But I don't think that's the real reason. Could someone shed some light on this?
He want his curves to be geometrically reduced. Having multiple factors leads to reducible curves, which are then singular at the intersection points of the components. But having repeated factors leads to multiple components (i.e. if you think scheme-theoretically, then the zero locus of $f^2 = 0$ is a thickening of the curve $f = 0$). He wants to avoid this. (Although I don't know what particular later statements he has in mind that motivate this, most likely he wants to be able to apply the Nullstellensatz and related results, and conclude that $f$ is determined by its collection of zeroes over $\overline{k}$ --- a statement that would be false if $f$ was allowed to have multiple factors.)