On Unsolvable Equations of Prime Degree
Abstract
Leopold Kronecker observed that either all roots or only one root of a solvable irreducible equation of odd prime degree with integer coefficients are real. This gives a possibility to construct specific examples of equations not solvable by radicals. A relatively elementary proof without using the full power of Galois theory is due to Heinrich Weber. We give a rather short proof of Kronecker’s Theorem with an argument that is slightly different from Weber’s. Several modern presentations of Weber’s proof contain inaccuracies, which can be traced back to an error in the original proof. We discuss this error and how it can be corrected.
Fields of Study & Subjects
Algebraic and Geometric Analysis
advanced mathematical theories
Publication details
- Journal / source
- American Mathematical Monthly
- Publisher
- Informa UK Limited
- Volume
- 133
- Issue
- 7
- Pages
- 634–648
- Publication date
- 20 May 2026
- Language
- EN