polynomial analogon for Fermat’s last theorem

For polynomialsPlanetmathPlanetmath with complex coefficients, there is an analogon of Fermat’s last theorem.  It can be proven quite elementarily by using Mason’s theorem (1983), but the original proof (about in 1900) was based on methods of algebraic geometryMathworldPlanetmathPlanetmathPlanetmath.

Theorem.  For an integer n greater than 2, there exist no non-constant coprimeMathworldPlanetmath polynomials x(t), y(t), z(t) in the ring [t] satisfying

[x(t)]n+[y(t)]n=[z(t)]n. (1)

Remark.  For  n=2, the equation (1) is in e.g. as



  • 1 Serge Lang: “Die abc-Vermutung”.  – Elemente der Mathematik 48 (1993).
Title polynomial analogon for Fermat’s last theorem
Canonical name PolynomialAnalogonForFermatsLastTheorem
Date of creation 2013-03-22 19:13:05
Last modified on 2013-03-22 19:13:05
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 8
Author pahio (2872)
Entry type Theorem
Classification msc 12E05
Classification msc 11C08
Synonym Fermat’s last theorem for polynomials
Related topic MasonsTheorem
Related topic WeierstrassSubstitutionFormulas