Power Residue Criteria for Quadratic Units and the Negative Pell Equation

http://dx.doi.org/10.4153/CMB-2003-004-1
Canad. Math. Bull. 46(2003), 39-53

  Published:2003-03-01
 Printed: Mar 2003

Let $d>1$ be a square-free integer. Power residue criteria for the fundamental unit $\varepsilon_d$ of the real quadratic fields $\QQ (\sqrt{d})$ modulo a prime $p$ (for certain $d$ and $p$) are proved by means of class field theory. These results will then be interpreted as criteria for the solvability of the negative Pell equation $x^2 - dp^2 y^2 = -1$. The most important solvability criterion deals with all $d$ for which $\QQ (\sqrt{-d})$ has an elementary abelian 2-class group and $p\equiv 5\pmod{8}$ or $p\equiv 9\pmod{16}$.