The successive minima in the geometry of numbers and the distinction between algebraic and transcendental numbers

This paper discusses an application of Minkowski's theory of the successive minima in the geometry of numbers to the problem of the approximation of an algebraic or transcendental number a by algebraic numbers. I consider for simplicity only real numbers a. However, it is obvious that an analogous theory can be established for complex numbers, and also for p-adic numbers, as well as for the field of formal ascending or descending Laurent series with coefficients in an arbitrary field.