The theory of functions of several variables turns out to be essentially more difficult than the theory of one variable because of the existence of p… - Carl Ludwig Siegel

I am afraid that mathematics will perish by the end of this century if the present trend for senseless abstraction — as I call it: theory of the empty set — cannot be blocked up.

Let θ be an algebraic integer and assume that all conjugates of θ, except θ itself, have an absolute value less than 1. Then –θ also has this property; on the other hand, θ is real. Without loss of generality, we may therefore suppose θ ≥ 0. Since the norm of θ is a rational integer, we have θ ≥ 1, except for the trivial case θ = 0. Recently, R. Salem ... discovered the interesting theorem that the set S of all θ is closed and that θ = 1 is an isolated point of S. Consequently there exists a smallest θ = θ₁ > 1. We shall prove that θ₁ is the positive zero of x³ – x – 1 and that also θ₁ is isolated in S. Moreover we shall prove that the next number of S, namely the smallest θ = θ₂ > θ₁, is the positive zero of x⁴ – x³ – 1 and that θ₂ is again an isolated point of S. Since θ₁ = 1.324..., θ₂ = 1.380..., both numbers are less than 2^½; therefore our statements are contained in the following: . Let θ be an algebraic integer whose conjugates lie in the interior of the unit circle; if ±θ ≠ 0, 1, θ₁, θ₂, then θ² > 2.

One of the many importants ideas introduced by Minkowski into the study of convex bodies was that of gauge function. Roughly, the gauge function is the equation of a convex body. Minkowski showed that the gauge function could be defined in a purely geometric way and that it must have certain properties analogous to those possessed by the distance of a point from the origin. He also showed that conversely given any function possessing these properties, there exists a convex body with the given function as its gauge function.