SWhat changed
Lindemann adapted Hermite's exponential-integral arithmetic to prove that π is not a root of any nonzero polynomial with rational coefficients. The proof uses symmetry among algebraic conjugates, integer or algebraic-integer relations that become too small, and a nonzero determinant; the application combines the exponential relation e^(iπ)=−1 with the algebraic hypothesis. Transcendence excludes exact squaring of a circle by a finite construction using only a compass and unmarked straightedge.
Scientific scope
Lindemann adapted Hermite's exponential-integral arithmetic to prove that π is not a root of any nonzero polynomial with rational coefficients. The proof uses symmetry among algebraic conjugates, integer or algebraic-integer relations that become too small, and a nonzero determinant; the application combines the exponential relation e^(iπ)=−1 with the algebraic hypothesis. Transcendence excludes exact squaring of a circle by a finite construction using only a compass and unmarked straightedge. The admitted result is transcendence of π and its classical Euclidean constructibility consequence, not simply irrationality, a finite numerical test or an approximation algorithm. Rational coefficients mean the usual rational-number algebraic notion; historical real-or-complex rational wording is not arbitrary complex coefficients. Hermite's e theorem is antecedent, not this new result. Section4states broader exponential-transcendence/linear-independence results with proofs expressly reserved for fuller later treatment; those are not promoted here to a fully proved Lindemann–Weierstrass theorem or a claim of algebraic independence of e and π. Other construction tools and approximate quadratures are not ruled out.
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 admitted result is transcendence of π and its classical Euclidean constructibility consequence, not simply irrationality, a finite numerical test or an approximation algorithm. Rational coefficients mean the usual rational-number algebraic notion; historical real-or-complex rational wording is not arbitrary complex coefficients. Hermite's e theorem is antecedent, not this new result. Section4states broader exponential-transcendence/linear-independence results with proofs expressly reserved for fuller later treatment; those are not promoted here to a fully proved Lindemann–Weierstrass theorem or a claim of algebraic independence of e and π. Other construction tools and approximate quadratures are not ruled out.
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.
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