[my formulation:]
> > "... Jacobi often claimed to be in possession of results on quintic
> > residuacity, but refused to share any details even when asked to do
> > so."
>
> At first sight this seems to me to be an overstatement, but I would
> be very glad to know further of these "often" claims. Which are the
> authoritative sources to the point? ...
I wrote that years ago with the background of Jacobi's priority
claims against Eisenstein in mind and before I learned about the
contents of Jacobi's 1837-lectures on cyclotomy. Jacobi had given
the first proofs of cubic and quartic reciprocity in his lectures
at Koenigsberg (1837) but never published them. What he published
in 1837 was a paper on applications of cyclotomy without proofs.
When Eisenstein's proofs appeared in the mid 1840's, Jacobi
republished his 1837-article in Crelle and added the (in)famous
footnote claiming that "These proofs which were known for several
years ... have lately been published by Dr. Eisenstein". In a
private letter (very likely infuriated by Gauss's praises of
Eisenstein), he even went so far as to call Eisenstein a liar
(see Eisenstein's Coll. Papers).
Jacobi's claims concerning quintic reciprocity occur in his
paper in Crelle (1839) as well as in the French translation
published in 1843. When Reuschle asked him in 1846 for
criteria concerning the character of (10/p)_n for n = 5, 7, 8, 9,
Jacobi sent the answer for n=8 and remarked that criteria for
(10/p)_5 would depend on the factorization of p in \Z[\zeta_5].
The last remark is not much of an answer, and criteria for
the octic character of 10 _could_ have been found by "induction".
Anyway, today I think that Jacobi knew how to prove Eisenstein's
quintic reciprocity law; he certainly had the necessary
background, as his 1837 lectures and a yet unpublished
manuscript (proving some claims made in his 1837 paper
on Gauss sums) show. In the current version of my manuscript,
the formulation is as follows:
Some remarks in Jacobi's note [1839] suggest that Jacobi
was in possession of the "Eisenstein reciprocity law"
for quintic and octic residues as early as 1839. [...]
Jacobi did not publish any details, however.
I think that's ok now.
franz