I have been studying the proof of Fermat’s last theorem for regular primes featured in Fermat's Last Theorem : A Genetic Introduction to Algebraic Number Theory (Edwards). This proof of Kummer’s theorem was appealing to me because it doesn’t need class field theory nor p-adic methods, but there is one step that bugs me. On page 240, the author uses the congruence
$ζ^k(1-σ(ζ))^{x_1}(1-σ^2(ζ))^{x_2}\dots (1-σ^μ(ζ))^{x_μ}\equiv c\mod (p)$
(here $σ(ζ)=ζ^γ$ where γ is a generator of $\mathbb Z/p\mathbb Z$), to justify that a certain rational fraction can be written as a sum of certain polynomials with integers coefficients. Precisely, letting
$f(t)=t^k(1-σ(t))^{x_1}(1-σ^2(t))^{x_2}\dots (1-σ^μ(t))^{x_μ}$,
the author writes
$f(t)=c+pq(t)+(1+t+\dots +t^{p-1})r(t)$
where $q,r\in \mathbb Z[X]$.
This looks odd to me, because some of the $x_i$ are negative. I understand how this works if we know that this fraction is actually a polynomial, but I can’t explain why it must be. I tried to solve this issue by considering a polynomial which coincides with f at ζ (we know that it is an algebraic integer). But we also need the derivatives to match at ζ, to obtain an equality between logarithmic derivatives. It is easy to find a polynomial with rational coefficients which coincides with the fraction and its derivative at ζ. But to adapt the rest of the proof, i would like the coefficients of this polynomial to have positive p-adic valuations, and I am not certain this is possible.