Presumably anyone who is on this forum is well aware of the famousFermat's Last Theorem(FLT): "There exists no solutions to the equation $\displaystyle a^n+b^n=c^n$ for $\displaystyle n>2$, $\displaystyle a,b,c\in\mathbb{Z}$ and $\displaystyle abc\ne0$." Moreover, I'm sure it's equally well known that this 'theorem' was really a conjecture for over three hundred years until Sir Andrew Wiles of Princeton finally proved a result which implied FLT in 1994-1995 in one of the most stupendous mathematical feats in the history of the subject (the proof may be found here for those interested).

Despite the infamous difficulty of FLT most students of mathematics can attest that they have, at one point or another, devoted a day, week, or month to trying to prove it--after all, it doesn't lookthathard. Well, a more mathematically cultured person may have a different pursuit, namely asking whether an analogue of FLT holds for other fields (rings). Namely, given a ring $\displaystyle R$ does there exist non-trivial solutions to $\displaystyle a^n+b^n=c^n$ for $\displaystyle n>2$? Of course, this isn't true at all for general rings or fields (take $\displaystyle \mathbb{C}$)! So perhaps the question needs to be narrowed from a general ring to a more specific, tame ring, the question is, which ring? Well, besides $\displaystyle \mathbb{Z}$ perhaps every math students favorite ring is $\displaystyle k[x]$ (the ring of polynomials with coefficients in $\displaystyle k$, for those unfamiliar to algebra) for some field $\displaystyle k$, so why not this? I mean, for the sake of simplicity why not restrict to the easiest infinite field $\displaystyle \mathbb{C}$? So, the question now is, does there exist solutions to $\displaystyle a(x)^n+b(x)^n=c(x)^n$ for coprime $\displaystyle a(x),b(x),c(x)\in\mathbb{C}[x]$ for $\displaystyle n>2$? There is clearly two distinct cases to this problem, namely when $\displaystyle a(x),b(x),c(x)$ are constant, and when they are all non-constant (clearly these are the only two cases). The first of these cases, as discussed, is evidently false, and so the crux of the problems occurs when one decides to consider only when $\displaystyle \min\{\deg(a(x)),\deg(b(x)),\deg(c(x))\}\geqslant 1$. So, what's the answer, do there exist non-trivial solutions?

I won't keep you in suspense, the answer is no. So, who won a Fields medal for the proof? Which great mathematician solved this problem? Well, shockingly the person who solved this originally didn't win a Fields medal, isn't famous, and isn't even (at least widely) known. In fact, one can prove this theorem (more generally over fields of characteristic zero) with such elementary methods that a mathematically inclined high school student would have no difficulty understanding it. Particularly, once one proves the so-called Mason-Stothers Theorem (which states that under the conditions of our 'theorem' one has that $\displaystyle \max\{\deg(a(x)),\deg(b(x)),\deg(c(x))\}+1 \leqslant \text{number of complex roots of }a(x)b(x)c(x)$) then one can prove the FLT for $\displaystyle \mathbb{C}[x]-\mathbb{C}$ in less than a page (a proof of both the Mason-Stothers and the FLT for $\displaystyle \mathbb{C}[x]-\mathbb{C}$ here).

So my question, the discussion I'd like to start is, what intuition is there for this version of the FLT to be SO much easier than for $\displaystyle \mathbb{Z}$?