Inspired by [this question](https://mathoverflow.net/questions/150835/does-fermats-last-theorem-hold-in-the-ordinals) and its answer, I am curious whether or not Fermat's last theorem holds in the Grothendieck ring of the ordinals under Hessenberg (commutative) operations.


The excellent answer to the other question by Emil Jeřábek uses the fact that the sum of the coefficients of an ordinals Cantor normal form is $0$ iff the ordinal itself is $0$, which is no longer true in the Grothendieck ring. The 'Cantor normal forms' now have integer coefficients instead of natural number coefficients, so for example $\omega-1$ is nonzero with coefficients summing to $0$.



Does Fermat's last theorem hold in the Grothendieck ring of the ordinals?
It does hold. Assume for contradiction that $x^n+y^n=z^n$ is a solution with $x,y,z$ nonzero, and put $w=xyz$. It suffices to find a ring homomorphism $f$ to $\mathbb Z$ such that $f(w)$ is nonzero, as then $f(x)^n+f(y)^n=f(z)^n$ is a nontrivial solution in $\mathbb Z$ contradicting the Fermat–Wiles theorem.


Now, your ring is isomorphic to the ring of multivariate integer polynomials with the indeterminates being ordinals of the form $\omega^{\omega^\alpha}$. We thus get an evaluation homomorphism to $\mathbb Z$ whenever we fix all the indeterminates to some integers.


So, it suffices to show that a nonzero integer multivariate polynomial (the $w$ above) does not compute a constant zero function on $\mathbb Z$. This follows by induction on the number of variables, using the fact that a nonzero univariate polynomial can only have finitely many integer roots.


