If a non-real logarithm λ and its conjugate are algebraically dependent, e^{|λ|} is transcendental

The modulus of a logarithm, in transcendence degree one.

Let $\lambda$ be a non-real logarithm of an algebraic number, and suppose that $\bar\lambda$ is algebraic over $\mathbb{Q}[\lambda]$. Since $\lambda$ is transcendental, this is the same as saying that $\lambda$ and $\bar\lambda$ are algebraically dependent. Then

is transcendental.

This is the homogeneous companion of Diaz's modulus conjecture. The conjecture asks that $|\lambda|$ never be algebraic; this node says that $|\lambda|$ is not a logarithm of an algebraic number, whenever $\lambda$ and $\bar\lambda$ generate a field of transcendence degree one. At $\lambda = i\pi$ it gives the transcendence of $e^{\pi}$. The case $\lambda = \log 2 + i\pi$ is DiazModulus.exp_abs_log_two_add_i_pi_transcendental.

Proof. The matrix $\begin{pmatrix} |\lambda| & \bar\lambda \\ \lambda & |\lambda| \end{pmatrix}$ has determinant $|\lambda|^{2} - \lambda\bar\lambda = 0$. If $e^{|\lambda|}$ were algebraic, its entries would be non-zero logarithms of algebraic numbers, all algebraic over $\mathbb{Q}[\lambda]$, so by DiazModulus.four_exponentials_trdeg_one its rows or its columns would be $\mathbb{Q}$-linearly dependent. Either way $\lambda$ would be a rational multiple of the real number $|\lambda|$, against $\operatorname{Im}\lambda \neq 0$.

Attribution. Known: Proposition 2 of G. Diaz, J. Théor. Nombres Bordeaux 9 (1997), and Corollaire 7.4 of D. Roy and M. Waldschmidt, Ann. Sci. École Norm. Sup. (4) 30 (1997), found independently the same year; see also Exercise 15.16(c) of Waldschmidt's Diophantine Approximation on Linear Algebraic Groups (2000). The node Diaz.log_modulus_forces_independence states it with the four exponentials theorem as an explicit hypothesis; this node is unconditional. The contribution of this node is the formal proof.