If Σ β_k ℓ_k is algebraic for a ℚ-basis (β_k) of a number field and logarithms ℓ_k, every ℓ_k is 0

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If Σ β_k ℓ_k is algebraic for a ℚ-basis (β_k) of a number field and logarithms ℓ_k, every ℓ_k is 0 · Prove2Me