Solubility of Fermat's Equation with Exponent n over Q(zeta_n)


To all number theorists!

For any natural number n, let S(n) denote the following
statement:

"Fermat's equation x^n + y^n = z^n has only trivial solu-
tions (such that xyz = 0) over the cyclotomic field Q(zeta_n)."
(Here zeta_n is a primitive n th root of unity.)

I would like to learn your opinion about the following
question: Is it a well-founded conjecture that statement
S(n) holds for every n>2 ?

Here are the reasons why this conjecture seems to have a
considerable degree of plausibility (A)-E)):

A) Statement S(3) and S(4) have been proven by Gauss.

B) Proofs of both S(8) and S(9) are contained in my paper
"Solubility of the Equation x^q + y^q = z^q over Cyclo-
tomic Fields Q(zeta_n) for Some Small Values of q and n"
printed in "Monatshefte für Mathematik" 145, 207-227 (2005).

C) The validity of S(12) is implied by the fact that Fer-
mat's cubic equation x^3 + y^3 = z^3 has only trivial so-
lutions over Q(zeta_12). This can easily be deduced by
calculating the rank of a certain elliptic curve, or by using
elementary results contained in the last-mentioned paper.

D) Statements S(5), S(6), S(7), S(10), S(11), S(13), S(14)
are covered by Kummer's work and Hilbert's results ("Zahl-
bericht", 1897), which imply that Fermat's equation of
degree n has only trivial solutions over Q(zeta_n) if n
is a regular prime or a double of a regular prime. Thus,
we can conclude that S(n) is valid for 2 < n < 15.

E) For all n>14, it is again clear that S(n) holds if n
is a regular prime, or if n is a double of such prime.
What about other non-prime numbers n? According to
Debarre-Klassen's conjecture, for every non-prime num-
ber n it is a reasonable hypothesis that Fermat's
equation of degree n has only trivial solutions over
Q(zeta_n), due to the field degree being less than n-1.

Hence, can we argue that statement S(n) is conjecturally
valid for every n>2 ? What is the present knowledge about
the case of n being an irregular prime?

Best regards,
Peter Findeisen

.