From Abel to Kronecker: Episodes from 19th Century Algebra

Birgit Petri and Norbert Schappacher

A chapter in The Legacy of Niels Henrik Abel, 2004, pp 227-266 from Springer

Abstract: Abstract This paper is about Leopold Kronecker reading Niels Henrik Abel’s results and ideas on the resolution of algebraic equations. When the young Kronecker began to work on algebraic equations, he went back and forth between Abel’s works and his own ideas. And throughout his career he continued to position himself much nearer to Abel than to Galois. At the same time, his own creativity transformed Abel’s results and questions into something more arithmetic and fairly different. For instance, already in his very first publication on algebraic equations [41], when unfolding Abel’s problems on solvable equations, Kronecker essentially claimed both what is known today as the ‘Theorem of Kronecker and Weber,’ to the effect that every abelian extension of Q is cyclotomic, and its analogue for abelian extensions of $$ Q(\sqrt { - 1} ) $$ , and even indicated further generalizations.

Keywords: Elliptic Function; Galois Group; Solvable Equation; Fractional Linear Transformation; Abelian Extension (search for similar items in EconPapers)
Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-18908-1_7

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DOI: 10.1007/978-3-642-18908-1_7

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