In Part I, Mahler introduces his classification of complex numbers and the following two results are proved. Let be algebraic numbers that are linearly independent over the rationals and let be a Liouville number. Then, the numbers are algebraically independent over the field of algebraic numbers. Let be the real logarithm of a positive rational number not equal to one and let be a Liouville number. Then, and are algebraically independent over the field of algebraic numbers.
Part II continues the study of the same title by giving various bounds on polynomials evaluated at logarithms and exponentials. A new proof of the transcendence of is given as an application.
Reprint of the author's papers [J. Reine Angew. Math. 166, 118--136 (1931; Zbl 0003.15101; JFM 57.0242.03); ibid. 166, 137--150 (1932; Zbl 0003.38805; JFM 58.0207.01)].