Far from being able to even begin to answer the question, let
me make a few remarks that perhaps shed some light on the situation.
Gauss's result on the cubic character of $2$ was generalized in
Dedekind's highly underrated article on pure cubic fields
[Über die Anzahl der Idealklassen in reinen kubischen
Zahlkörpern; J. Reine Angew. Math. 121 (1900), 40-123]:
already in the early 1870s Dedekind had written
Let $k$ denote a rational integer whose cube root is irrational.
Then the equation $x^3 = k$ defines a pure cubic number field whose
discriminant has the form $D = -3g^2$, where the integer $g$ can
easily be determined from $k$. Consider all primes $p$ of the form
$3n+1$ that do not divide $k$ and modulo which the given integer $k$
is a cubic residue. With the help of the reciprocity law we then find
the following interesting result, which essentially was known already
to Gauss (and may be extended to general cubic fields): there are three
kinds of primitive binary quadratic forms $ax^2 + bxy + cy^2$ with
$D = b^2 - 4ac$ and representing different classes: the forms of the
first kind are a group whose forms represent exactly those prime numbers
modulo which $k$ is cubic residue.
He could not generalize his results, as promised, to general cubics,
although he correctly conjectured that this should be possible using
the theory of complex multiplication, i.e., class field theory for
complex quadratic number fields.
Dedekind also quotes a remark by Gauss, in which he determined the
cubic character of $5$ using binary quadratic forms (which coincides
with Satgé's example except that Gauss considered only forms
with even middle coefficient, which means that he has four forms
$(1,0,675)$, $(25,0,27)$, $(13,2,52)$, $(4,2,169)$ where Satgé
can do with one $(1,1,169)$.
Dedekind's result was generalized by Takagi [Sur les corps résolubles
algébriquement, C. R. Acad. Sci. Paris 171 (1920), 1202-1205].
Takagi presents Dedekind's theorem in the following form:
Let $D$ denote the discriminant of a cubic number field $k$;
then the class number of primitive quadratic forms with
discriminant $D$ is a multiple of $3$. A third of the classes
forms a group which can be characterized by the property that
the prime numbers not dividing $D$ and for which $D$ is a
quadratic residue, and which split into three different factors
in $k$, and only those, are represented by a quadratic form
in this group.
Then he proves his generalization:
Let $k$ be a solvable field of prime degree and $K_0$ the
corresponding cyclic field. If the ideal classes in $K_0$ are
defined modulo $f$, the class group contains a subgroup of
index $\ell$, which is characterized by the property that among
the prime numbers not dividing the discriminant of $k$ and
splitting completely in $K_0$, exactly those primes that split
completely in $k$ are the norms of ideals in $K_0$ lying in
the subgroup above.
If $k$ is a pure cubic field, then $K_0$ is the field of cube roots
of unity, and Takagi gets back Dedekind's result. For the proof,
Takagi used his class field theory.
Special cases of this result were rediscovered e.g. by
- D. Liu [Dihedral polynomial congruences and binary quadratic forms
in her Ph.D. thesis (Carleton 1992) supervised by
- Spearman & Williams
[The cubic congruence $x^3 + Ax^2 + Bx + C \equiv 0 \pmod p$
and binary quadratic forms,
J. London Math. Soc. (2) 46 (1992), no. 3, 397-410],
[The cubic congruence $x^3 + Ax^2 + Bx + C \equiv 0 \pmod p$
and binary quadratic forms II,
J. London Math. Soc. (2) 64 (2001), no. 2, 273-274].
See also
- D. Bernardi [ Résidus de puissances,
Semin. Delange-Pisot-Poitou 1977/78, Fasc. 2, Exp. No. 28, 12 pp.
The results by
- Weinberger [The cubic character of quadratic units,
Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder 1972, 241-242]
and the more recent ones by
- Sun [Cubic residues and binary quadratic forms,
J. Number Theory (2006)]
also follow this pattern (there are in fact a lot more articles
dealing with describing the splitting of primes in ring class fields
of quadratic number fields using binary quadratic forms).
Satgé considers the following situation: let $K$ be a normal
extension of the rationals with Galois group $G$, let $H$ be a
normal abelian subgroup, and let $k$ be the fixed field of $H$.
By applying class field theory to the abelian extension $K/k$, he
characterizes the decomposition law in $K$ by representation of
the primes in question by the norm forms attached to $k$, assuming
certain conditions on $H$. The norm forms have degree $(G:H)$;
thus the case where $H = 1$ is absolutely trivial since in this case
the splitting of primes in $K$ (for unramified primes in normal
extensions, all we need to know is the inertia degree) is described
by norms from $K$.
Overall these are all "abelian" phenomena and follow readily from
(abelian) class field theory. But of course it is legitimate to
ask how these fit into the nonabelian Langlands conjectures, since
after all we are dealing with nonabelian extensions here.
