The prominent mathematician Leopold Kronecker (1823 – 1891) is often relegated to footnotes and mainly remembered for his strict philosophical position on the foundation of mathematics. He held that only the natural numbers are intuitive, thus the only basis for all mathematical objects. In fact, Kronecker developed a complete school of thought on mathematical foundations and wrote many significant algebraic works, but his enigmatic writing style led to his historical marginalization. In 1887, Kronecker published an extended version of his paper, “On the Concept of Number,” translated into English in 2010 for the first time by Edward T. Dean, who confirms that Kronecker is “notoriously difficult to read.” In his paper, Kronecker proves that a so-called “algebraic number”, meaning any root of a polynomial with integer coefficients, can be isolated from the other roots of that polynomial, as Dean says, “using solely talk of natural numbers.” To ease the reader’s comprehension of Kronecker’s prose, here we explicate in detail the argument contained in that paper.
Dr. Richard Delaware, Teaching Professor, Department of Mathematics and Statistics, University of Missouri — Kansas City
Schneider, Richard B.
"Irrational Philosophy? Kronecker's Constructive Philosophy and Finding the Real Roots of a Polynomial,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 22
, Article 1.
Available at: https://scholar.rose-hulman.edu/rhumj/vol22/iss1/1