An algebraic number in the Q̄-span of the logarithms of algebraic numbers is 0 (Baker)

Here $\mathcal{L}$ is the set of logarithms of algebraic numbers, and $\widetilde{\mathcal{L}}$ is the $\overline{\mathbb{Q}}$-vector space spanned by $1$ and $\mathcal{L}$. If $z$ is algebraic and lies in the $\overline{\mathbb{Q}}$-span of $\mathcal{L}$, then $z = 0$. Equivalently $\overline{\mathbb{Q}} \cap \overline{\mathbb{Q}}\,\mathcal{L} = \{0\}$, so $\widetilde{\mathcal{L}} = \overline{\mathbb{Q}} \oplus \overline{\mathbb{Q}}\,\mathcal{L}$.

Proof. Choose a $\mathbb{Q}$-linearly independent $B \subseteq \mathcal{L}$ with the same $\mathbb{Q}$-span as $\mathcal{L}$; then $z = \sum c_\lambda\lambda$ over finitely many $\lambda \in B$ with $c_\lambda$ algebraic. If $z \neq 0$, Baker's theorem (Schanuel.baker_linear_forms_in_logarithms) with constant term $-z$ gives $-z + \sum c_\lambda\lambda \neq 0$, while this sum is $0$.

Novelty. Not asserted: a direct consequence of Baker's theorem (1966).