Hilbert's Seventh Problem: Solutions and
extensions

In the seventh of his celebrated twenty-three problems of 1900, David Hilbert proposed that mathematicians attempt to establish the transcendence of an algebraic number to an irrational, algebraic power. Partial solutions to this problem were given by A. O. Gelfond in 1929, R. O. Kuzmin in 1930, and K. Boehle in 1933. In 1934 the complete solution was obtained independently by A. O. Gelfond and by Th. Schneider. The partial solutions were based on ideas reminiscent of the nineteenth proofs for the transcendence of e (Hermite, 1873) and for the transcendence of pi (Lindemann, 1882); both of the general solutions relied on a new idea that opened the way for the development of a theory of transcendental numbers. In this course we will very briefly look at nineteenth century transcendence methods, examine the partial and complete solutions to Hilbert's seventh problem, and then study three ways in which mathematicians have generalized the theorem of Gelfond and Schneider. Our examination of these three generalizations will illustrate several of the novel ideas that played significant roles in twentieth-century transcendental number theory and allow us to offer several open questions. Prerequisites: Introductory courses in functions of a complex variable and abstract algebra. |