The canonical bundle of a genus-one curve is trivial

(Set-up.) By [F4] the canonical sheaf ${\omega }_C={\mathrm{\Omega }}_{C\mathrm{/}k}^1$ is invertible and ${\omega }_C\cong {\mathcal{O}}_C(K_C)$ for the canonical divisor $K_C$ of any nonzero rational differential, and the canonical divisors form a single linear equivalence class; by [F1] one has ${\mathrm{deg}}_k(K_C)=2g-2=0$, and by [F2] one has $h^0(C,{\omega }_C)=l(K_C)=g=1$.

The degree of the invertible sheaf ${\omega }_C$ is computed by the degree of any associated divisor, so $\mathrm{deg}({\omega }_C)={\mathrm{deg}}_k(K_C)=0$ by step 1.1.

By [F2] the space $H^0(C,{\omega }_C)$ has dimension one, hence is nonzero.

Since ${\omega }_C$ is an invertible sheaf of degree zero with nonzero $H^0$ by steps 2.1 and 2.2, the criterion [F3] gives ${\omega }_C\cong {\mathcal{O}}_C$.

(Canonical divisors are principal.) By step 3.1 ${\omega }_C$ is trivial, so a generator ${\omega }_0\in H^0(C,{\omega }_C)$, which exists because $h^0(C,{\omega }_C)=1$ by step 1.1, is a nowhere-vanishing global section; under an isomorphism ${\omega }_C\cong {\mathcal{O}}_C$ from step 3.1 it corresponds to a nonzero global section of ${\mathcal{O}}_C$, and the divisor of a nonzero global section of an invertible sheaf is effective of degree $\mathrm{deg}({\omega }_C)=0$ by [F5], hence is the zero divisor because the only effective divisor of degree zero on $C$ is $0$; thus the canonical divisor $\mathrm{div}({\omega }_0)=0$ is principal. Since by [F4] every canonical divisor is linearly equivalent to this one, every canonical divisor is the divisor of a rational function, that is, principal.

Assertions: ${\omega }_C\cong {\mathcal{O}}_C$ is step 3.1, $h^0(C,{\omega }_C)=1$ is step 1.1 (in particular no rational point of $C$ was used anywhere), and the principality of all canonical divisors is step 4.1; the Axiom of Choice [F6] is used exactly through the duality suppliers cited above.