Cantor's List of Real Algebraic Numbers of Heights 1 to 7

TL;DR

This paper systematically lists real algebraic numbers of heights 1 to 7, expanding on Cantor's original work by providing explicit enumerations for polynomials of various degrees.

Contribution

It provides a comprehensive enumeration of real algebraic numbers for heights 1 to 7 across different polynomial degrees, extending Cantor's initial results.

Findings

Explicit lists of algebraic numbers for heights 1 to 7

Systematic enumeration across polynomial degrees

Extension of Cantor's original countability proof

Abstract

Cantor gave in his fundamental article an elegant proof of the countability of real algebraic numbers based on a positive integer height, denoted by him as N, of integer and irreducible polynomials of given degree (denoted by him as n) with relative prime coefficients. The finite number of real algebraic numbers with given height he called phi(N), and gave the first three instances.\pn Here we give a systematic list for the real algebraic numbers of height, which we denote by n, for n from 1 to 7 and polynomials of degree k.