SWhat changed
Hermite constructed simultaneous rational approximations to exponentials and used their arithmetic structure to show that e satisfies no nonzero polynomial equation with integer, equivalently rational, coefficients. His treatment converts a putative integer-coefficient relation among distinct integral powers of e into integer linear forms that tend to zero. A nonvanishing determinant prevents all such forms from vanishing, yielding a contradiction.
Scientific scope
Hermite constructed simultaneous rational approximations to exponentials and used their arithmetic structure to show that e satisfies no nonzero polynomial equation with integer, equivalently rational, coefficients. His treatment converts a putative integer-coefficient relation among distinct integral powers of e into integer linear forms that tend to zero. A nonvanishing determinant prevents all such forms from vanishing, yielding a contradiction. The assessed theorem is the transcendence of e, not algebraic independence of e and pi or the later general algebraic-exponent Lindemann–Weierstrass theorem. The offprint presents an asymptotic argument followed by a second determinant argument described as more rigorous; the latter’s factorial decay, integrality and distinct-root determinant conditions are the source of the bounded record. Liouville’s earlier transcendental-number constructions and partial results concerning e are antecedents, not this full result. The discovery year refers to four 1873 Comptes rendus notes; the exact ETH offprint is undated in its printed imprint and institutionally catalogued [1873].
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The assessed theorem is the transcendence of e, not algebraic independence of e and pi or the later general algebraic-exponent Lindemann–Weierstrass theorem. The offprint presents an asymptotic argument followed by a second determinant argument described as more rigorous; the latter’s factorial decay, integrality and distinct-root determinant conditions are the source of the bounded record. Liouville’s earlier transcendental-number constructions and partial results concerning e are antecedents, not this full result. The discovery year refers to four 1873 Comptes rendus notes; the exact ETH offprint is undated in its printed imprint and institutionally catalogued [1873].
Contributors
- The named authors in the research record · SCIENCE: Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
Provisional WRS assessment, edition 2026-10-23.3. Independent expert calibration is pending. This grade does not certify correctness or assign ownership.
Full scientific record · Browse contribution grades · Request a correction