I'm asked to prove the following fact:
On a complex torus $X$ every canonical divisor is principal and vice-versa.
At this moment I know only the basic properties of divisors and that, if $K$ is a canonical divisor on $X$, then $\operatorname{deg} (K) = 0$. Any help would be appreciate, thanks.
The easiest way is writing out an explicit canonical divisor. For example, if $z_1,\ldots,z_n$ are local coordinates of the torus $X=\mathbb{C}^n/\Lambda$ for some lattice $\Lambda$, then $dz_1\wedge\cdots\wedge dz_n$ is a well-defined $n$-form on $X$. The divisor it defines is obviously the zero divisor, since it is nowhere vanishing. The zero divisor, in turn, is equal to the divisor of a constant function, and you have what you are looking for.