Cubic residues and binary quadratic forms (Q877923)

One of the oldest results on higher reciprocity is Euler's observation that \(2\) is a cubic residue modulo a prime \(p \equiv 1 \bmod 3\) if and only if \(p\) is represented by the quadratic form \(x^2 + 27y^2\) of discriminant \(-27\cdot 2^2\). This conjecture was later proved by Gauss and generalized considerably by \textit{R. Dedekind} [J. Reine Angew. Math. 121, 40--123 (1899; JFM 30.0198.02), Ges. Math. Werke II, 340--353; (1900; JFM 30.0198.02)], who showed that for any integer \(m\) (not a cube) there is a subgroup \(H\) of index \(3\) in the class group of forms of discriminant \(-27m^2\) with the property that \(m\) is a cubic residue modulo a prime \(p \equiv 1 \bmod 3\) if and only if \(p\) is represented by a form in \(H\). Dedekind knew that \(m\) is a cubic residue modulo such a prime \(p\) if and only if \(p\) splits in the pure cubic extension \(\mathbb Q(\root 3\of {p})\), and conjectured that the result above generalizes to \textit{all} cubic fields. This conjecture was proved (and generalized from cubic to solvable extensions) by \textit{T. Takagi} [C. R. Acad. Sci. Paris 171, 1202--1205 (1920; JFM 47.0147.02)] using his newly developed class field theory; the part related to cubic extensions was rediscovered later by \textit{B. K. Spearman} and \textit{K. S. Williams} [J. Lond. Math. Soc. (2) 46, No. 3, 397--410 (1992; Zbl 0724.11002); ibid. 64, No. 2, 273--274 (2001; Zbl 1017.11017)]. Probably unaware of Dedekind's results and mainly motivated by Emma Lehmer's work on the residuacity of quadratic units, \textit{P. J. Weinberger} [Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder 1972, 241--242 (1972; Zbl 0321.12002)] showed that a unit in a quadratic number field with discriminant \(d\) is a cubic residue modulo a prime ideal \({\mathfrak p}\) if and only if the norm of \({\mathfrak p}\) is represented by a quadratic form in an explicitly given subgroup of index \(3\) in the class group of forms of discriminant \(-3df^2\), where \(f\) is a small power of \(3\). His article ends with the remark that ``the theorem may be proved by standard class field theory calculations''. In this article, the author gives very explicit versions of Dedekind's and Weinberger's results. His proofs are elementary and use nothing beyond cubic reciprocity (and the fact that there are infinitely many inert primes in cubic number fields); the price the author has to pay for avoiding class field theory is high, however: the proof of the main results [see also J. Number Theory 113, No. 1, 10--52 (2005; Zbl 1101.11003)] is highly technical. The author's results are too numerous to list here; a quite simple one is the following: for primes \(p \equiv 1 \bmod 4\) and \(p \equiv \pm 1 \bmod 3\) we have \((2+\sqrt{3}\,)^{(p \mp 1)/3} \equiv 1 \bmod p\) if any only if \(p\) is represented by \(x^2 + 81y^2\) or \(2x^2 + 2xy + 41y^2\) (these two forms form a subgroup of index \(3\) in the class group of forms of discriminant \(-4 \cdot 81\)). It is also shown how to apply these results to derive divisibility properties of Lucas sequences.