Kronecker’s arithmetical theory of algebraic quantities. (English) Zbl 0756.01025

The author points to some interesting topics in Kronecker’s works with the intention of leading more people to read the important but somewhat overshadowed nineteenth-century mathematician. Kronecker’s construction of the splitting field of a polynomial, “the heart of Galois theory”, is described in detail. Kronecker considered polynomials over what he termed “natürliche Rationalitätsbereiche” which the author expresses in modern terms as fields of quotients of the ring of polynomials in \(n\) indeterminates with integral coefficients. For a second topic the author selects Kronecker’s generalization to arbitrary number fields of Kummer’s theory of ideal prime factors for cyclotomic fields. Kronecker’s approach is contrasted with Dedekind’s for algebraic number fields; the latter, the author believes, “does not capture the essence of the matter in a way that generalizes to other cases”. He concludes with an account of Kronecker’s philosophy and of how Kronecker’s algorithmic approach in mathematics has been unjustly regarded in the history of mathematics as a failed approach of interest only for its oddity.


01A55 History of mathematics in the 19th century
11-03 History of number theory
01A70 Biographies, obituaries, personalia, bibliographies

Biographic References:

Kronecker, L.