On Unsolvable Equations of Prime Degree

Journal article, 2026

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.