Extension of a theorem of Cauchy and Jacobi (Q801117)

Cauchy (1840) and Jacobi (1846) determined the products \(\prod_{k}kf!\) modulo a prime \(p=qf+1\) \((q=a^ 2+b^ 2,\) a prime \(\equiv 1 (4)\), \(a\equiv 1 (4))\), where k runs through the quadratic residues and the quadratic non-residues mod q respectively, in terms of parameters in representations of \(p^ h\) or \(4p^ h\) by binary quadratic forms. The most striking feature of this business is the tie up between the exponent h and the class number of the related quadratic field. If only the above products of factorials were either more natural to look at in the first place or tied up with yet another facet of the subject, this tie up would be most exciting! In the present paper the authors determine four similar products of factorials modulo \(p=qf+1,\) \(q\equiv 5 (8)\), \(q>5\), (where k runs through the group S of quartic residues mod q and the 3 cosets of S respectively) in terms of parameters in representations of \(16p^ h\) by quaternary quadratic forms (cf. Dickson's solution to the cyclotomic problem for the modulus 5). This determination is far more complicated than the corresponding classical one. There is again this intriguing tie up between the exponent h and the class number of the imaginary quartic field \({\mathbb{Q}}(i\sqrt{2q+2a\sqrt{q}}),\quad q=a^ 2+b^ 2\equiv 5 (8),\) a odd, which the authors achieve by using a variety of clever means and complicated manipulations. Examples are given at the end and special cases of congruences for binomial coefficients are discussed.