Baker over a number-field basis for ℒ_u = ℒ + ℚu + ℚū (u transcendental over the field of ℒ̃, uū algebraic): if Σ βₖℓₖ is algebraic with ℓₖ ∈ ℒ_u, every ℓₖ is 0

Here $\mathcal{L}$ is the set of logarithms of algebraic numbers (the $z \in \mathbb{C}$ with $e^z$ algebraic), and $\widetilde{\mathcal{L}}$ is the $\overline{\mathbb{Q}}$-vector space spanned by $1$ and $\mathcal{L}$. Let $F$ be the subfield of $\mathbb{C}$ generated by $\widetilde{\mathcal{L}}$, that is, by $\overline{\mathbb{Q}}$ and $\mathcal{L}$. Let $u \in \mathbb{C}$ be transcendental over $F$, with $u\bar u$ algebraic, and put $\mathcal{L}_u = \mathrm{span}_{\mathbb{Q}}(\mathcal{L}) + \mathrm{span}_{\mathbb{Q}}(u, \bar u)$; as $\mathcal{L}$ is a $\mathbb{Q}$-vector space, this is $\mathcal{L} + \mathbb{Q}u + \mathbb{Q}\bar u$. Let $K \subseteq \mathbb{C}$ be a subfield with a finite basis $(\beta_k)_{k \in \iota}$ over $\mathbb{Q}$ (so $K$ is a number field), and let $\ell_k \in \mathcal{L}_u$ for $k \in \iota$. If $\sum_k\beta_k\ell_k$ is algebraic, then $\ell_k = 0$ for every $k$.

Proof. Write $\ell_k = c_k + p_ku + q_k\bar u$ with $c_k \in \mathcal{L}$ and $p_k, q_k \in \mathbb{Q}$ (DiazModulus.mem_logAlg_adjoin_circle_point_iff), and $a = \sum_k\beta_k\ell_k$. Then $(\sum_k\beta_kc_k - a) + (\sum_k\beta_kp_k)u + (\sum_k\beta_kq_k)\bar u = 0$, with every coefficient in $F$, which contains the algebraic numbers and $\mathcal{L}$; so DiazModulus.transcendental_add_mul_add_mul_conj forces $\sum_k\beta_kp_k = \sum_k\beta_kq_k = 0$, and as the $p_k, q_k$ are rational and $\beta$ is a $\mathbb{Q}$-basis, every $p_k$ and $q_k$ is $0$. Hence $\ell_k = c_k \in \mathcal{L}$, and Baker's theorem over a number-field basis (Transcendence.baker_number_field_basis, Theorem 4.5 of Waldschmidt's book) gives $\ell_k = 0$.

Novelty. With $\mathcal{L}_u$ in place of $\mathcal{L}$, this is Baker's theorem (1966) in the form of Theorem 4.5 of Waldschmidt's book (2000), from which it is derived; Baker's theorem enters through Transcendence.baker_number_field_basis, proved in this mission following Chapter 4 of that book. Not found in the sources read; routine.