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Consider the transcendental extension Q(t) of the field of rationals. To Q(t) adjoin the root of the polynomial x^5+t^5=1. The resulting field Q(t)[x] is a radical extension of Q(t). Is it true that the only solutions to the equation X^5+Y^5=1 in the field Q(t)[x] are (0,1), (1,0), (t,x), (x,t), (1/t,-x/t) and (-x/t, 1/t)?

Comment: Using the ABC theorem one can prove that the Fermat curve X^n+Y^n=1 does not have a non-trivial solution in Q(t) for n>2. In particular in Q(t) the equation X^5+Y^5=1 does not have non-trivial solutions.

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up vote 12 down vote accepted

First, replace Q by the complex numbers C.

Write A = C[x,y]/(x^n+y^n-1). Then the field you write down, call it K, is the fraction field of A, which is the function field of the Fermat Curve.

Finding a solution (X,Y) to x^n + y^n = 1 is equivalent to finding a map A --> K, where one sends x to X and y to Y. Any map to a field factors though A/P for some prime ideal P of A.

Since (x^n+y^n-1) is irreducible, the only primes in A are either (0) or maximal ideals m with A/m = C.

If A/P = C, then the map A --> K factors through C. Maps from A to C correspond to complex points on the Fermat curve.

If P = 0, then A --> K extends to a non-trivial map from K --> K. Giving a map between function fields is equivalent to giving a map of the corresponding smooth projective curves (in the opposite direction). Thus, the question becomes: what are the non-trivial maps from the Fermat curve to itself? For symmetry reasons, write the Fermat curve as x^n + y^n + z^n = 0 (over C).

Assume that n > 3. Since the genus of the Fermat curve is > 1, the Reimann-Hurwitz formula tells us that any non-trivial map must be an automorphism. On the other hand, the isomorphism group of the Fermat curve is the semi-direct product of (Z/nZ)^2 by S_3. (Explicitly, this corresponds to multiplying (x,y,z) by n-th roots of unity, and permuting the entries.) In terms of the affine coordinates the S_3 action is generated by (x,y)->(y,x) and (x,y)->(1/y,-x/y).

To summarize, the K points are given by the (finitely many) automorphisms, and the C-points of the curve. You have noted all the automorphisms over Q, I would note that you are also invoking the non-trivial fact that the Fermat curve has no non-trivial points over Q, which (even for n = 5) that is harder than everything else in this answer (and is especially harder for general n > 2!). (The "ABC theorem" for polynomials only tells you that there are no non-constant solutions in C[t]. In general, a solution in C[t] will correspond to a map from P^1-->X, which can be non-trivial only if X has genus 0.)

Note that this argument used basically nothing about the Fermat curve itself --- any curve of genus > 1 has only finitely many automorphisms.

If g = 1, then there are some non-trivial maps from a curve of genus g to itself, but Riemann-Hurwitz tells us that they are all unramfied. X/C in this case is an elliptic curve, so there will be infinitely many non-trivial solutions in K, corresponding to the rational functions giving the multiplication by [n] map (or more if X has CM). In the Fermat case, this means that for n = 3 there will be many more solutions.

If g = 0, there are buckets of maps from P^1 to itself, as one knows in the Fermat case when n = 2 or 1.

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Very nice indeed. – David Lehavi Nov 7 '09 at 8:29
it looks like a correct answer! Cant be 100% positive since I am new in the area and do not know the terminology and notation. see my next question extending this. – Bakh Nov 7 '09 at 18:41

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