= Freshman's dream
{wiki}
In a <commutative ring> of prime <characteristic> $p$, the <freshman's dream> identity is $(x+y)^p=x^p+y^p$. The <binomial theorem> proves it because $p$ divides every intermediate <binomial coefficient> $\binom pk$, $0<k<p$. Iterating gives $(x+y)^{p^m}=x^{p^m}+y^{p^m}$. For example, $(x+y)^3=x^3+y^3$ over $\mathbb F_3$, while over the <real numbers> the cross terms generally remain. Prime characteristic is essential to this ring-wide statement; an arbitrary positive composite characteristic is insufficient.
Back to article page
