Fermat-Goss-Denis theorem

Fermat's last theorem is the claim that has no solutions in non-zero integers for (see also Fermat last theorem). However, over a function field (cf. also Algebraic function), with of non-zero characteristic , the appropriate generalization is not just to take , and as polynomials over in . In any event, in characteristic zero, or for prime to the characteristic , it is fairly easy to see, by descent on the degrees of a putative solution , that there is not even a non-trivial solution over , with the algebraic closure of .

In 1982, D. Goss [a1] formulated a suitable analogue for the case . Goss notes that, traditionally, Fermat's equation is viewed as , where the connection with cyclotomic fields, and thence the classical exponential function, is displayed: the zeros of are precisely the roots of unity. But in characteristic the analogue of the exponential function comes by way of the Drinfel'd module; more specifically, the Carlitz module. A familiar and elementary manifestation of such things is the Hilbert theorem 90, whereby a cyclic extension of degree in characteristic is not given by a zero of , but of .

Let be the field of elements. The equation that in this context appears to raise issues analogous to those provoked by the classical Fermat equation is

where is the Carlitz module determined by and denotes the Frobenius mapping relative to , i.e. the mapping that gives powers. To say that is the Carlitz module is to require also that . Goss [a1] deals, à la Kummer, with the case of this equation when is a regular prime of .

The equation has two important parameters, the element of and the order . As usual, a solution with is called non-trivial. When , Goss shows that in analogy with the equation there are an infinity of solutions. Suppose is monic. L. Denis [a2] proves that if , and , there is no non-trivial solution. If , and , there is a unique solution proportional in to the triplet in the case , where is a square in ; and if , , then there is a solution only if is of the shape , and it is . Denis deals completely with the remaining cases . Because is -linear, one can now easily produce the results for not monic.

In settling the general case, Denis [a2] speaks of the Fermat–Goss theorem. It seems appropriate here to write of the Fermat–Goss–Denis theorem.

References

[a1]	D. Goss, "On a Fermat equation arising in the arithmetic theory of function fields" Math. Ann. , 261 (1982) pp. 269–286
[a2]	L. Denis, "Le théorème de Fermat–Goss" Trans. Amer. Math. Soc. , 343 (1994) pp. 713–726