The abc Conjecture and a Fermat's Last Theorem for Polynomials

Pierre de Fermat (1601-1665) stated his "Last Theorem'' (that xⁿ + yⁿ = zⁿ has no nontrivial solutions in Ζ for n ≥ 3) in the margin of his copy of Diophantus's Arithmetica in 1637. In one of the boldest claims by one of the brightest individuals in the history of mathematics, Fermat wrote that he had a proof, but that he did not have enough room to write it in the margin. It is very likely that his proof was incomplete. Nonetheless, his innocent enough statement incited hundreds of capable (and not so capable) individuals into feverish work for over three and one-half centuries. These individuals made great accomplishments in mathematics; the development of Modern Algebra being one of the foremost. This intriguing chapter in mathematics' history came to a close in 1995 with the work of Andrew Wiles and Richard Taylor.

We will present a proof of a version of Fermat's Last Theorem for polynomials. In developing the proof of this result, the important open Number Theory problem known as the abc Conjecture will be presented. Shinichi Mochizuki's work on the conjecture will be mentioned, but not addressed. And though the proof involves some topics from abstract algebra, the audience will be reminded of basic definitions. We note that the proof of the polynomial version of Fermat's Last Theorem is significantly shorter than the integer version.